Defendants Alberto Flaharty (“Alberto”), Tyrone Grimes, Carf Flaharty (“Carf’), Garth Bruce, and Lawrence Johnson appeal from judgments entered in the United States District Court for the Eastern District of New York following a jury trial before John Gleeson, Judge, convicting each defendant of conspiring between 1992 and April 1998 to distribute and possess with intent to distribute cocaine base (crack cocaine) and marijuana, in violation of 21 U.S.C. §§ 846 and 841(b)(1)(A) (count one), and of conspiring between September 1996 and April 1998 to do so within 1,000 feet of a school zone, in violation of 21 U.S.C. §§ 846 and 860 (count two); convicting Bruce, Carf, and Grimes of engaging in a continuing criminal enterprise, in violation of 21 U.S.C. *188§§ 848(a) and (c) (count three); convicting Johnson of unlicensed dealing in firearms, in violation of 18 U.S.C. §§ 922(a)(1)(A) and 924(a)(1)(D) (count four); and convicting Bruce (count five) and Grimes (count six) of possessing firearms as previously convicted felons, in violation of 18 U.S.C. §§ 922(g)(1) and 924(a)(2). Defendants were sentenced principally to the following prison terms: Alberto, two concurrent 180-month terms; Grimes, three concurrent 420-month terms and one 120-month term to be served concurrently with his 420-month terms; Carf, three concurrent life terms; Bruce, three concurrent life terms and one 60-month term to be served concurrently with his life terms; and Johnson, two concurrent life terms and one 60-month term to be served concurrently with his life terms. Grimes’s imprisonment is to be followed by a total of five years of supervised release; each other defendant’s imprisonment is to be followed by a total of 10 years of supervised release.
On appeal, defendants advance numerous contentions, including challenges to the indictment’s sufficiency to charge a continuing criminal enterprise, to the court’s evidentiary rulings and instructions to the jury, to the sufficiency of the evidence to support their convictions on certain counts, and to the validity of their convictions and sentences in light of Apprendi v. New Jersey, 530 U.S. 466, 120 S.Ct. 2348, 147 L.Ed.2d 435 (2000). In addition, Bruce, Carf, and Grimes contend that they could not properly be convicted of both engaging in a continuing criminal enterprise and participating in a narcotics conspiracy that is a lesser included offense of the continuing criminal enterprise. The government properly concedes the correctness of this lesser-included-offense contention, see Rutledge v. United States, 517 U.S. 292, 300, 307, 116 S.Ct. 1241, 134 L.Ed.2d 419 (1996), and we therefore reverse the convictions of Bruce, Carf, and Grimes on count one of the superseding indictment and remand to the district court for the dismissal of,that count against those defendants. Finding no basis for reversal in any of defendants’ other contentions, we affirm the judgments of conviction in all other respects.
I. BACKGROUND
The present appeals arise out of the prosecution of the five appellants and others in connection with their participation in a narcotics distribution enterprise on Beach 26th Street in Far Rockaway, New York. At trial, the government presented the testimony of more than 30 witnesses, including, some 20 local and federal law enforcement officers; coconspirators Dave Hamilton and Kyheem Butler, respectively a leader and a seller; and persons who purchased drugs from the coconspirators for resale elsewhere or for personal use. The government also presented numerous exhibits, including guns, ammunition, bags of crack cocaine, and other drug paraphernalia. Taken in the light most favorable to the government, the evidence revealed the following.
A. The Beach 26th Street Conspiracy Operations
In approximately 1992, Hamilton, Bruce, and a pair known as “Spanky” and “Brian” (“Spanky/Brian”) entered into a time-sharing arrangement with respect to drug sales on the Beach 26th Street block, with each of the three principals, in turn, controlling the block for a two-day period. The principals employed workers to act as sellers and lookouts during their respective turns. In 1993, the principals modified the agreement so that each principal would control the block’s drug trade for a week at a time rather than two days at a time; *189this arrangement lasted from 1993 until mid-1995.
During the latter period, Carf served as a lieutenant for Hamilton and Bruce and was responsible for managing the day-today affairs of the business, including distributing drugs to the sellers, collecting the money from the sales, and paying the sellers for their work. Butler worked as a lookout for Hamilton and Bruce; Carfs brother Alberto was a seller or a lookout for Hamilton and Bruce. Hamilton and Bruce each sold a weekly average of 10-18 ounces of crack cocaine and 5-8 pounds of marijuana. ■ The operation’s weekly gross income averaged $14,000-$16,000 and was as high as $20,000.
In mid-1995, Spanky/Brian left the block, and Hamilton assumed control of drug sales for two of every three weeks, with Bruce retaining control of the third week. This alignment lasted until late 1997. Carf, except for a period between May and September 1995 during which he was incarcerated, continued to serve as lieutenant for both Hamilton and Bruce. At various times, Grimes, Alberto, and Butler worked for Hamilton and Bruce as sellers. The operation sold an average of nine ounces of crack cocaine a week and grossed about $8,000 to $12,000 a week.
In September 1996, a public elementary school opened on Beach 29th Street, less than 500 feet from Beach 26th Street. As a result, Hamilton sought to distance himself physically from the Beach 26th Street operation. He left New York City for several weeks in 1996-1997, during which Carf assumed control of his weeks. In late 1997, Hamilton moved to Ohio; Grimes, who had become a lieutenant for Hamilton and Bruce in mid 1997, took over Hamilton’s branch of the operation, controlling two out of every three weeks until March 1998. During Grimes’s period of control, he averaged weekly sales of approximately 4 — 4/6 ounces of crack cocaine, grossing about $3,000 to $4,000 a week. Alberto sold for both Grimes and Bruce and employed others to sell for him. Bruce, during that period, grossed approximately $2,000 to $2,500 a week.
Johnson, who lived near the border between Virginia and Tennessee, supplied the coconspirators with weapons. He purchased guns mostly from flea markets around his home, brought the guns to New York, and resold them in Far Rockaway. Shannon Hale, Johnson’s confidante who had accompanied him on a few such trips, had helped him load the car with guns, and had witnessed gun sales in New York, testified that Johnson said he had driven to New York nearly every week between 1993 and 1995, transporting 9-14 guns on each trip. Johnson sold the guns for cash, crack cocaine, or powder cocaine, and then resold the drugs in the Virginia-Tennessee area; from his gun and drug sales, he grossed $30,000 to $40,000 a week. Johnson sold at least four guns to Hamilton, sold at least one each to Bruce, Carf, and Grimes, and sold guns to other members of the enterprise as well.
B. The Present Prosecution
In 1994-1998, local law enforcement officers conducted surveillances, undercover buys, and buy-and-bust operations in investigating drug sales in the Beach 26th Street area and acquired evidence of narcotics sales by Carf, Grimes, Alberto, and others, and gun sales by Johnson. Execution of search warrants in 1996-1998 on apartments occupied by Bruce and Alberto, respectively, and on a bungalow used by Hamilton and Bruce, turned up bags containing crack cocaine, as well as an abundance of drug-related paraphernalia including triple-beam scales, narcotics packaging materials, handguns, ammuni*190tion, empty gun boxes, pagers, and walkie-talkies.
A multi-agency, multi-jurisdiction investigation of the conspiracy concluded in May 1998, and federal indictments were returned against numerous individuals. The present defendants were eventually tried on a six-count superseding indictment charging each of them with two counts of narcotics conspiracy, charging Bruce, Carf, and Grimes with conducting a continuing criminal enterprise, and charging Bruce, Grimes, and Johnson with firearms offenses, all as indicated above.
Each defendant was convicted on all counts in which he was charged. On a special verdict form that posed successive sets of questions as to each individual defendant, the jury found, inter alia, “that the government has proved beyond a reasonable doubt that the amount of crack cocaine [each defendant] conspired to distribute and possess with intent to distribute equaled at least fifty (50) grams.” Defendants were sentenced as indicated above, and these appeals followed.
II. DISCUSSION
On appeal, defendants make numerous arguments, either directly or by joining the applicable arguments of their code-fendants. They contend principally that the trial court improperly curtailed their cross-examination of Hamilton, gave the jury erroneous instructions as to the evidence needed to return a verdict of guilty on count two, i.e., the charge of conspiracy to distribute drugs near the school zone, and that the court was not authorized to impose enhanced punishment in connection with that count. They also contend that their narcotics conspiracy convictions and sentences violate the principle set out in Apprendi. In addition, Bruce, Carf, and Grimes challenge the indictment’s sufficiency to charge a continuing criminal enterprise; Bruce contends that the government relied on inadmissible and insufficient evidence to show that he had substantial earnings from that enterprise. Johnson principally challenges the sufficiency of the evidence to show his participation in any narcotics conspiracy and contends that his sentence is excessive. Grimes contends that the government’s summation as to the guns’ nexus with interstate commerce relied on evidence outside the record. For the reasons that follow, we find no basis for reversal in these contentions or in any others except, as indicated above, the Rutledge challenge by Bruce, Carf, and Grimes to their convictions on the count-one conspiracy charge.
A. Limitation on Cross-Examination of Hamilton
Much of the evidence as to the internal workings of the Beach 26th Street drug operation was provided through the testimony of Hamilton, who had been a leader of the conspiracy from approximately 1992 until at least late 1997. Hamilton, inter alia, identified numerous members of the conspiracy in addition to the present defendants, described their roles at various points in the conspiracy, gave estimates as to the quantities of narcotics sold and the respective incomes produced by his and Bruce’s branches of the operation, and recounted his and his coconspirators’ gun purchases from Johnson. Defendants contend that their right to confront Hamilton was infringed when the trial court refused to allow them to inquire about Hamilton’s involvement in the murder of one Roland Evans. We disagree.
The trial court has wide discretion to impose limitations on the cross-examination of witnesses. See, e.g., Delaware v. Van Arsdall, 475 U.S. 673, 679, 106 S.Ct. 1431, 89 L.Ed.2d 674 (1986); United *191States v. Griffith, 284 F.3d 338, 352 (2d Cir.2002). Under Federal Rule of Evidence 608(b), the district court may restrict cross-examination about specific instances of prior conduct if it finds that the conduct is not probative of truthfulness. Further, under Rule 403, the district court may exclude even relevant evidence if it finds that the “probative value [of the testimony] is substantially outweighed by the danger of unfair prejudice, confusion of the issues, or misleading the jury, or by considerations of undue delay, waste of time, or needless presentation of cumulative evidence.” Fed.R.Evid. 403. Even where the court has inappropriately excluded evidence under Rule 608(b), reversal is not required if “ ‘the jury was already in possession of sufficient information to make a discriminating appraisal of the particular witness’s possible motives for testifying falsely in favor of the government.’ ” United States v. Tillem, 906 F.2d 814, 827-28 (2d Cir.1990) (quoting United States v. Singh, 628 F.2d 758, 763 (2d Cir.), cert. denied, 449 U.S. 1034, 101 S.Ct. 609, 66 L.Ed.2d 496 (1980)).
In the present case, the trial court determined both that the Evans murder was not relevant to Hamilton’s credibility and that there was ample other material through which defendants could test his credibility and expose any bias. We see no error or abuse of discretion in this ruling. Murder generally is not a crime of dishonesty, and nothing about the Evans murder suggested that it would in any way reflect on Hamilton’s truthfulness. Further, the record supports the court’s finding that there was abundant other infor-mátion available for the cross-examination of Hamilton, and indeed, defendants questioned him extensively with respect to his numerous false statements to law enforcement officers and the numerous acts of criminal conduct that he had admitted in connection with his cooperation agreement with the government. The latter included his leadership of the Beach 26th Street drug conspiracy, his violent armed robberies of other drug dealers, his efforts to cover up his prior crimes, and his participation in another murder. In sum, there was ample grist for cross-examination, and the exclusion of evidence as to Hamilton’s participation in the murder of Evans provides no basis for reversal.
B. Challenges to the School-Zone Violation Convictions
1. The Supplemental Jury Charge
Count one of the superseding indictment charged defendants with drug conspiracy from 1992 to 1998. Count two charged that their conspiracy involved distribution within 1,000 feet of a school from September 1996 to 1998. During its deliberations, the jury sent a note to the court, asking whether a defendant who was guilty of participating in the count-one conspiracy could be convicted on count two even if there was no evidence of that defendant’s participation after September 1996. In response, the district court referred the jury to a previous written instruction that had been given on the issue of single versus multiple conspiracies and then stated:
I instruct you in order to find the defendant guilty of Count Two, it’s not necessary that the government prove beyond a reasonable doubt that the defendant actively participated in that conspiracy after September of 1996 as long as the government has proven beyond a reasonable doubt that the defendant you are considering joined that conspiracy, the one alleged in Count Two, even if he did so before September of '96. The long accepted rule is that a conspirator is presumed to continue in the conspiracy until the last overt act of any of his co-conspirators.
*192The court added that that presumption may be overcome when a defendant produces affirmative evidence of his withdrawal from the conspiracy, and it instructed the jury on the ingredients of such a withdrawal.
Defendants challenge this supplemental instruction, arguing that it allowed the jury to convict them on count two simply on the basis of their guilt on count one. We see no error in the charge.
It is well established that where the government has shown that a conspiracy existed and that a given defendant was a member of it, his membership is presumed to continue until the last overt act by any of the coconspirators, unless the defendant proves that the conspiracy was terminated or that he took affirmative steps to withdraw. See, e.g., Hyde v. United States, 225 U.S. 347, 369-70, 32 S.Ct. 793, 56 L.Ed. 1114 (1912); United States v. Panebianco, 543 F.2d 447, 453 (2d Cir.1976), cert. denied, 429 U.S. 1103, 97 S.Ct. 1129, 51 L.Ed.2d 553 (1977). Withdrawal is an affirmative defense, see, e.g., United States v. Eisen, 974 F.2d 246, 268 (2d Cir.1992), cert. denied, 507 U.S. 1029, 113 S.Ct. 1840, 123 L.Ed.2d 467 (1993), and the defendant has the burden of showing that he performed affirmative acts that were "inconsistent with the object of the conspiracy and communicated in a manner reasonably calculated to reach co-conspirators," United States v. United States Gypsum Co., 438 U.S. 422, 464, 98 S.Ct. 2864, 57 L.Ed.2d 854 (1978); mere cessation of activity is not sufficient, see, e.g., United States v. Eisen, 974 F.2d at 268. Positive evidence of withdrawal is required in order to provide assurance that the defendant genuinely removed himself from the conspiracy and is not simply attempting an after-the-fact escape from liability. See, e.g., United States v. Greenfield, 44 F.3d 1141, 1150 (2d Cir.1995).
In the present case, none of the defendants except Carf, see Part II.B.2. below, argued that he had presented any evidence from which it could be inferred that he had withdrawn from the conspiracy. Indeed, the three defendants who testified at trial stated that they had never been participants in the first place. For example, Carf testified that although he had sold drugs on Beach 26th Street between 1992 and 1994, he had done so for himself and not as part of any conspiracy. Johnson denied ever selling drugs or transporting firearms to New York. And Grimes testified, inter alia, that he had known nothing about the Beach 26th Street drug operation, had never participated in any organized drug selling on the block, and never knew that Hamilton controlled drug sales on Beach 26th Street.
The jury plainly found that each defendant had participated in the count-one conspiracy. In the absence of any evidence of withdrawal, the court correctly instructed that a defendant's continued participation could be presumed and that the government was not required to present evidence of further affirmative acts by a nonwith-drawing defendant.
2. Carfs Claim of Withdrawal Through Incarceration
Nor do we find merit in Carl's contention that he is entitled to a new trial because the jury was not properly instructed as to the factors it should consider in determining whether his participation in the conspiracy ended with his 1995 incarceration. "[WlIhile arrest or incarceration may constitute a withdrawal from a conspiracy, it does not follow that in every instance it must," United States v. Agueci, 310 F.2d 817, 839 (2d Cir.1962) (emphases in original), cert. denied, 372 U.S. 959, 83 *193S.Ct. 1013, 10 L.Ed.2d 11 (1963), and whether a coconspirator’s imprisonment constitutes a withdrawal “must be decided by the jury in light of the length and location of the internment, the nature of the conspiracy, and any other available evidence,” United States v. Panebianco, 543 F.2d at 454 n. 5. “A conspirator whq presents evidence of his imprisonment during the course of the conspiracy is entitled to a jury instruction on withdrawal.” United States v. Salameh, 152 F.3d 88, 150 (2d Cir.1998) (per curiam), cert. denied, 525 U.S. 1112, 119 S.Ct. 885, 142 L.Ed.2d 785 (1999). However, the failure to give such an instruction is not a basis for reversal unless the defendant can show that a properly instructed jury would likely have found that he withdrew from the conspiracy. See id.
Carf has made no such showing. The conspiracy endured for some six or seven years. Carf was incarcerated nearby and for only a few months, from May to September of 1995. Prior to his incarceration, he played an integral role in the conspiracy, serving as a lieutenant for both Hamilton and Bruce. Although Carf testified that he thereafter worked as a truck driver and did not sell drugs, the government presented evidence that after his release from jail he in fact resumed, and perhaps even elevated, his participation in the conspiracy-. For example, Joel Lipsky, a customer, testified that Carf participated in a sale of crack cocaine to Lipsky in November 1996. Butler, a street-seller beginning in 1996, testified that Carf was a lieutenant in 1996, replenishing Butler’s drug supplies and paying him; Hamilton testified that Carf remained one of his lieutenants until at least mid-1997. And Hamilton testified that Carf took charge of several of Hamilton’s allotted weeks in 1996-1997. Given the ample evidence of Carfs participation in the conspiracy after his September 1995 release from jail, together with the relative brevity of his incarceration, Carf cannot demonstrate that the jury, if properly instructed, would likely have found that he withdrew from the conspiracy-
3. The Punishment Imposed for the School-Zone Violation
Defendants also challenge the enhanced penalties imposed on them pursuant to 21 U.S.C. § 860 for conviction of conspiracy to distribute and possess with intent to distribute narcotics within 1,000 feet of a school zone. They argue, citing United States v. Ekinci 101 F.3d 838 (2d Cir.1996), that the penalties set out in § 860(a) are not applicable unless there is a substantive violation of § 841(a), whereas defendants were convicted only of conspiracy to violate § 841(a). We are unpersuaded.
Section 860 describes an offense whose pertinent elements are (a) the performance of certain acts that are prohibited by 21 U.S.C. § 841(a)(1), and (b) the proximity of those acts to a school, and it penalizes such an offense severely:
Any person who violates section 841(a)(1) ... of this title by distributing, possessing with intent to distribute, or manufacturing a controlled substance in or on, or within one thousand feet of, the real property comprising a public or private elementary, vocational, or secondary school ... is (except as provided in subsection (b) of this section) subject to (1) twice the maximum punishment authorized by section 841(b) of this title; and (2) at least twice any term of supervised release authorized by section 841(b) of this title for a first offense.
21 U.S.C. § 860(a). Subsection (b) of § 860 provides penalties of even greater severity for second and successive § 860(a) offenders. In United States v. Ekinci, the § 860 count of the indictment charged that *194the defendant physician, within 1,000 feet of a school, had distributed, possessed with intent to distribute, and dispensed a controlled substance in violation of § 841(a)(1). The jury found him guilty, but its verdict was ambiguous as to which of those acts-distributing, possessing, or dispensing-it found he had performed. We noted that the § 841(a) actions referred to in § 860 include distributing and possessing with intent to distribute, but do not include dispensing; and we concluded that the enhanced penalty provided by § 860 could not be applied because the jury might have found the defendant guilty only of dispensing. See United States v. Ekinci, 101 F.3d at 839-41.
Ekinci is not applicable here, for count two of the superseding indictment in the present case did not mention dispensing. Count two here mentioned only conspiracy to “distribute” and to “possess with intent to distribute,” both of which acts are listed in both §§ 841(a) and 860. Hence the verdict in the present case was not ambiguous, for the jury could not have found that a defendant performed an act that was prohibited in § 841(a) but not in § 860. In sum, unlike in Ekinci, the guilty verdict did not leave open the' possibility that a defendant had engaged in conduct for which a § 860 sentence enhancement is not authorized.
Nor was the sentence unauthorized simply because the defendants were not convicted of substantive offenses. The conspiracy section of Title 21 provides that
[a]ny person who attempts or conspires to commit any offense defined in this subchapter shall be subject to the same penalties as those prescribed for the offense, the commission of which was the object of the attempt or conspiracy.
21 U.S.C. § 846. Since count two of the superseding indictment alleged that the object of defendants’ conspiracy was distribution of, and possession with intent to distribute, narcotics within 1,000 feet of an elementary school in violation of § 860, the penalty for the conspiracy alleged in count two was the same as the penalty prescribed for a substantive violation of § 860. We see no error in the imposition of the § 860 penalties.
C. The Apprendi Challenges and the Jury Interrogatories
In sentencing defendants on counts one and two, the district court applied 21 U.S.C. § 841 (b)(l)(A)(iii), which provides, to the extent pertinent here, that a defendant convicted of distributing or possessing with intent to distribute 50 or more grams of crack cocaine “shall be sentenced to a term of imprisonment which may not be less than 10 years or more than life,” 21 U.S.C. § 841(b)(1)(A)(iii); see 21 U.S.C. § 846 (penalties for § 846 conspiracy are the same as those prescribed for the substantive offense that was the object of the conspiracy). As indicated above, Alberto was sentenced to concurrent 180-month terms; Grimes was sentenced to concurrent 420-month terms; Bruce, Carf, and Johnson were sentenced to concurrent life terms. Defendants do not challenge the sufficiency of the evidence that the quantity of crack cocaine in which they conspired to deal was at least 50 grams. (Nor could they. Hamilton testified that they distributed 9-18 ounces a week through late 1997; at the end of the conspiracy period, they were distributing 4-4/6 ounces a week. Given that an ounce is the equivalent of more than 28 grams, Hamilton’s testimony indicated that the coconspirators sold more than 100 grams every week, and in some weeks sold more than 500 grams.) Instead, defendants contend that these sentences violate the principle of Apprendi v. New Jersey, 530 U.S. at 490, 120 S.Ct. 2348 (“Other than the fact of a prior conviction, any fact that increases the penalty *195for a crime beyond the prescribed statutory maximum must be submitted to a jury, and proved beyond a reasonable doubt.”), because drug quantity was not alleged in the superseding indictment or found by the jury beyond a reasonable doubt. We disagree.
This case was tried in the spring of 1999, more than a year before Apprendi was decided, but just weeks after the Supreme Court decided Jones v. United States, 526 U.S. 227, 119 S.Ct. 1215, 143 L.Ed.2d 311 (1999). In Jones, the Court ruled that facts that increase the maximum statutory penalty under the federal carjacking statute, 18 U.S.C. § 2119, are elements of that offense, not mere sentencing factors, and thus must be charged in the indictment, submitted to the jury, and proven beyond a reasonable doubt. Jones, 526 U.S. at 239, 252, 119 S.Ct. 1215. In light of Jones, the district court in the present case sua sponte distributed to the parties, on the day before the government rested, a proposed verdict sheet that included conspiracy-count interrogatories focusing on drug quantity. These interrogatories, for each defendant, read as follows:
1. Count One: Conspiracy to Distribute and to Possess with Intent to Distribute Cocaine and Marijuana
a. How do you find [this defendant]?
Guilty_Not guilty_
b. If you find [this defendant] guilty of the conspiracy alleged in Count One, do you find that the government has proved beyond a reasonable doubt that the amount of crack cocaine [this defendant] conspired to distribute and possess with intent to distribute equaled at least fifty (50) grams?
Proved_Not proved_
2. Count Two: Conspiracy to Distribute and to Possess with Intent to Distribute Cocaine and Marijuana Within One Thousand (1000) Feet of a Public Elementary School
a. How do you find [this defendant]?
Guilty_Not guilty_
b. If you find [this defendant] guilty of the conspiracy alleged in Count Two, do you find that the government has proved beyond a reasonable doubt that the amount of crack cocaine [this defendant] conspired to distribute and possess with intent to distribute within one thousand (1000) feet of a public elementary school equaled at least fifty (50) grams?
Proved_Not proved_
The jury found each defendant guilty on both conspiracy counts. And it found that each defendant’s membership in a conspiracy to deal in at least 50 grams of crack cocaine had been proven beyond a reasonable doubt.
Although defendants correctly point out that there was no allegation of drug quantity in the superseding indictment, they made no such' objection in the district court, and their objection on appeal is thus reviewable only under the plain-error standard, see Fed.R.Crim.P. 52(b). Under that standard, a defendant must demonstrate that (1) there was error, (2) the error was “plain,” (3) the error prejudicially affected his “substantial rights,” and (4) the error “seriously affect[ed] the fairness, integrity or public reputation of judicial proceedings.” United States v. Olano, 507 U.S. 725, 732, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993). “An error affects substantial rights if it is prejudicial and it affected the outcome of the district court proceedings.” United States v. McLean, 287 F.3d 127, 135 (2d Cir.2002) (internal quotation marks omitted); see also United *196States v. Thomas, 274 F.3d 655, 666-67 (2d Cir.2001) (en banc) (applying plain-error analysis to claim of Apprendi error).
Defendants have not met this standard. Although the superseding indictment omitted any allegation as to drug quantity, that omission cannot have caused any defendant to believe that quantity was not a factor in the case. While defendants might not have realized at the outset that the issue of quantity would be decided by the jury, it was well established that the district court, in determining facts material to sentencing, may rely on evidence admitted at trial, see, e.g., United States v. Miller, 116 F.3d 641, 685 (2d Cir.1997), cert. denied, 524 U.S. 905, 118 S.Ct. 2063, 141 L.Ed.2d 140 (1998); United States v. Carmona, 873 F.2d 569, 574 (2d Cir.1989); and the quantity issue clearly was material to sentencing both under the statute, which prescribes minimum and maximum prison terms depending in part on quantity, see, e.g., 21 U.S.C. § 841(bXl)(A) (minimum of 10 years, maximum of life for a violation involving 50 or more grams of crack cocaine); Id. § 841(b)(1)(B) (minimum of five years, maximum of 40 years for a violation involving five or more but less than 50 grams of crack cocaine); id. § 841(b)(1)(C) (maximum of 20 years for an indeterminate quantity of crack cocaine), and under the Sentencing Guidelines, see, e.g., Guidelines § 2D1.1(c) (drug table) (base offense level for a given type of narcotics determined by quantity). Thus, every defendant had an obvious incentive from the outset to challenge any quantity evidence the government presented at trial.
We see no error whatever in the district court's decision to submit interrog-atones to the jury with respect to drug quantity. "Th~ decision to use interrogatories to determine quantity, as with their use in general, lies in the discretion of the district judge," United States v. Madkour, 930 F.2d 234, 237 (2d Cir.), cert. denied, 502 U.S. 911, 112 S.Ct. 308, 116 L.Ed.2d 251 (1991), and given the then-recent holding in Jones, it was surely prudent for the court in this case to have the jury determine drug quantity and to do so under the beyond-a-reasonable-doubt standard.
As the quantity issue was actually submitted to the jury and was decided beyond a reasonable doubt, we cannot conclude that the omission in the superseding indictment prejudicially affected any defendant's substantial rights or seriously affected the fairness, integrity, or public reputation of these proceedings.
Although defendants also now contend that there were procedural errors in the submission of the interrogatories to the jury, they made no such objections at trial. The proposed verdict sheet was given to the parties before the government rested its case. After the conclusion of all the evidence, the district court gave the defendants an opportunity to object. No defendant made any objection either to the giving of such interrogatories or to the language used. No defendant suggested that he had had an insufficient opportunity to review the verdict sheet or to contest the issue of narcotics quantity. Again applying the plain-error standard, we see no procedural error that prejudicially affected any defendant's substantial rights or that in any way adversely affected the fairness, integrity, or public reputation of the proceedings.
Finally, we note also that Alberto would have no possible claim under Apprendi even if the jury had not made any finding with respect to quantity, because the maximum prison term for a defendant convicted of trafficking in an indeterminate quantity of cocaine base is 20 years, or 240 months, see 21 U.S.C. § 841(b)(1)(C); see *197also United States v. Thomas, 274 F.3d at 660 n. 2. Alberto was sentenced to two concurrent terms of 180 months, well below that statutory maximum.
D. Challenges to the Continuing Criminal Enterprise Convictions
Count three of the superseding indictment charged Bruce, Carf, and Grimes with
engag[ing] in a continuing criminal enterprise, in that they committed felony violations of Title 21, United States Code, Sections 841(a)(1) and 846, which violations were part of a continuing series of violations of those statutes undertaken by the above-referenced defendants in concert with five or more other persons with respect to whom the above-referenced defendants occupied positions of organizer, supervisor and manager, and from which continuing series of violations the above-referenced defendants obtained substantial income and resources.
Bruce, Carf, and Grimes contend that the indictment was deficient because it failed to specify the violations that constituted the "series" necessary for a conviction of engaging in a continuing criminal enterprise ("CCE"). In addition, Bruce challenges the sufficiency and admissibility of the evidence to show that he received substantial income from the enterprise. We reject all of these challenges.
1. Challenges to the Sufficiency of the CCE Indictment
A continuing criminal enterprise, as proscribed by 21 U.S.C. § 848, is defined in part as "a continuing series" of felony drug violations of any of the provisions in the subchapters comprising §~ 801-971 of Title 21. See 21 U.S.C. § 848(c). We have interpreted "a continuing series" to, mean at least three felony drug violations committed over a definite period of time. See, e.g., United States v. Aiello, 864 F.2d 257, 264 (2d Cir.1988).
In Richardson v. United States, 526 U.S. 813, 119 S.Ct. 1707, 143 L.Ed.2d 985 (1999), the Supreme Court held that in order to find a defendant guilty of engaging in a continuing criminal enterprise, the jury may not simply agree that the defendant committed three underlying crimes but must unanimously agree on which of the (three or more) individual violations constituted the continuing series. See id. at 824, 119 S.Ct. 1707. Although Richardson requires that the jury be unanimous on each of the constituent felonies, we have held that an indictment that does not identify which of many alleged felonies constituted the series is not thereby defective. In Santana-Madera v. United States, 260 F.3d 133 (2d Cir.2001) ("Santana-Madera"), cert. denied, - U.S. -, 122 S.Ct. 817, 151 L.Ed.2d 701 (2002), we noted the lack of specificity in the indictment:
[L]ike nearly all pre-Richardsom CCE indictments, Santana-Madera's indictment did not specify which offenses constituted the "continuing series of violations." The CCE count simply alleged that Santana-Madera organized or managed at least five other persons in connection with three or more of the nine federal drug law violations alleged in the indictment.
260 F.3d at 136. However, we rejected the contention that the lack of specificity meant that the CCE count fai1~d to charge an offense. We concluded that in "al-leg[ing] that Santana-Madera supervised five or more people in the commission of three or more of the nine violations of the drug laws listed in the indictment[, tlhe indictment thus alleged all the elements of the CCE offense, ~.. even after Richardson." Santana-Madera, 260 F.3d at 140 n. 3.
*198Count three of the superseding indictment in the present case was less informative than the indictment in Santana-Madera as to the felonies constituting the series, for count three referred only to "felony violations of Sections 841(a)(1) and 846," and counts one and two simply charged § 846 conspiracies to violate §~ 841(a)(1) and 860; but we cannot conclude that the superseding indictment thereby failed to allege an offense. In order to state an offense, "[a]n indictment need only track the language of the statute and, if necessary to apprise the defendant `of the nature of the accusation against him,' state time and place in approximate terms." United States v. Bagaric, 706 F.2d 42, 61 (2d Cir.) (quoting Russell v. United States, 369 U.S. 749, 766, 82 S.Ct. 1038, 8 L.Ed.2d 240 (1962)), cert. denied, 464 U.S. 840, 104 S.Ct. 134, 78 L.Ed.2d 128 (1983), abrogated on other grounds by National Organization for Women, Inc. v. Scheidler, 510 U.S. 249, 114 S.Ct. 798, 127 L.Ed.2d 99 (1994). Count three here closely tracked the language of § 848(c), and it alleged that the continuing series of felonies were violations of §~ 841(a)(1) and 846, and that the enterprise was conducted "fun or about and between 1992 and April 1998 ... within the Eastern District of New York and elsewhere." We thus conclude that count three did not fail to charge an offense.
Finally, we note that the jury was properly instructed, in accordance with Richardson, that it must be unanimous as to each constituent felony in order to find defendants guilty of continuing criminal enterprise:
I must emphasize that if you do find beyond a reasonable doubt that the defendant did commit a series of drug crimes as claimed by the government, you must unanimously agree among yourselves on at least three of the acts in the series in order to find [the continuing series of violations] element of the CCE count to be established. In short, as I mentioned before, it can't be the case that some of you agree as to some, others of you agree as to others. You must all unanimously agree beyond a reasonable doubt as to three or more narcotics related offenses before you can find the defendant you are considering guilty of the CCE count charged in Count Three.
In sum, we see no lack of notice in the superseding indictment and no error in the submission of the CCE count to the jury.
2. Bruce's Challenge to the Evidence of His Income
Bruce also challenges the admissibility and the sufficiency of the government's evidence to show that he derived substantial income from the enterprise, as required by 21 U.S.C. § 848(c)(2)(B). His admissibility challenge is meritless; his sufficiency challenge is frivolous.
Bruce argues that the government's evidence of his income was inadmissible because it consisted of impermissible hearsay and lay opinion. He concedes that "these objections are largely unpre-served"; but he argues that the admission of the testimony constituted plain error. (Bruce brief on appeal at 25.) We are unpersuaded.
Virtually all of the evidence as to the level of Bruce's earnings during the conspiracy period came from the testimony of Hamilton. Hamilton was present in Far Rockaway during most of the conspiratorial period, was essentially Bruce's partner in the drug enterprise even though they divided the weeks, had personal knowledge of the quantities of drugs being sold and the money grossed from those sales, and testified based on that personal knowledge. It was well within the bounds of discretion *199for the court to allow Hamilton to estimate the amount of income received by the respective coconspirators. See generally Fed.R.Evid. 701; United States v. Rivera, 22 F.3d 430, 434-35 (2d Cir.1994); Brady v. Chemical Construction Corp., 740 F.2d 195, 201 (2d Cir.1984).
Hamilton testified that from mid-1993 through 1997, Bruce controlled one out of every three weeks on the block, selling crack cocaine and marijuana every day of his week, 24 hours a day. During the earlier part of that period, Hamilton likewise controlled one week, and in then-respective weeks, Hamilton and Bruce sold an average of some 10-18 ounces of crack cocaine and 5-8 pounds of marijuana. Hamilton’s own average weekly gross intake was about $14,000 to $16,000 and was sometimes as high as $20,000; and from a gross of between $16,000 and $20,000, Hamilton’s profit was between $8,000 and $12,000. Hamilton testified that from mid 1993 to 1995, Bruce’s average gross income was about $15,000 a week.
During the latter part of the conspiracy period, Hamilton controlled two of every three weeks, while Bruce still controlled one week. They each continued to sell 24 hours a day, every day during their respective weeks. Hamilton testified that from 1995 until late 1997, he and Bruce averaged weekly sales of about nine ounces of crack cocaine and grossed about $8,000 to $12,000 a week. Bruce interposed no objection to Hamilton’s testimony as to Bruce’s income prior to the middle of 1996, and that testimony indicated that just from 1993 through mid 1996, Bruce, in the weeks he controlled, grossed at least $598,000. Bruce’s contention that the evidence was insufficient to show that he received substantial income from the enterprise is thus frivolous, even without considering his sales from mid-1996 to April 1998.
Hamilton testified that beginning in the summer of 1996, he was not always present to oversee his Beach 26th Street sales, and in late 1997 he left New York. Grimes took over Hamilton’s weeks; but Hamilton retained an interest in the level of sales in the Beach 26th Street operation, for Hamilton' (and his mother) shared in Grimes’s profits. Hamilton testified that Grimes sold about 4-4]é ounces of crack cocaine a week and grossed about $3,000 to $4,000. When Hamilton was asked how much Bruce grossed during that period, Bruce made his first.objection to Hamilton’s testimony. The court sustained the objection, instructing the government that it needed to lay a proper foundation. Hamilton then provided a foundation, testifying, without further objection by Bruce, that he had knowledge as to the amount Bruce was making in an> average week during the 1997-98 period. He stated that he knew because he “usually talked to them” and “was told” that Bruce was making approximately $2,000 to $2,500 a week. Bruce now argues that this testimony was hearsay.
In order to meet the plain-error standard with respect to this contention, Bruce must demonstrate that the district court committed an error in admitting Hamilton’s testimony, that the error was plain, that it prejudicially affected Bruce’s substantial rights, and that it seriously affected the fairness of the trial. See United States v. Olano, 507 U.S. at 732, 113 S.Ct. 1770. He has not met any part of this standard.
First, it is not clear that Hamilton’s testimony that, as to the 1997-98 period, “the[y]” told him how much money Bruce was earning was hearsay. Obviously the statement was objectionable for lack of specificity; but no objection was made, and Hamilton was not asked to clarify, who “the[y]” were. It is possible that one per*200son to whom Hamilton referred was Bruce himself; to the extent that Hamilton’s testimony repeated what Bruce had told him, it would not have been hearsay because the statement of a party, when offered against him, is defined as nonhearsay, see Fed.R.Evid. 801(d)(2)(A). It is also possible that “the[y]” included Alberto, Carf, or any of several other coconspirators who worked for both Hamilton and Bruce, and that the court, if the issue had been raised, would have found that the statements were made in furtherance of the conspiracy. To that extent, those statements would have been nonhearsay as defined by Fed. R.Evid. 801(d)(2)(E). Thus, we cannot say that the admission of Hamilton’s testimony as to Bruce’s income was error; or that, if error, the error was plain; or that it prejudiced Bruce’s substantial rights. Further, in light of the facts (a) that Bruce refrained from making the obvious objection to Hamilton’s reference to “them” and sought no clarification of that reference, and (b) that a clarification might well have been adverse to Bruce’s interests by showing that “the[y]” were none other than Bruce and his agents and coconspirators, we cannot conclude that the admission of that testimony adversely affected the fairness of the trial.
In sum, the government proved that Bruce received substantial income from the continuing criminal enterprise, and its reliance on the testimony of Hamilton to prove that element provides no basis for reversal.
E. Challenges by Johnson
Johnson attempts to distance himself from the other defendants in making several challenges to his conviction and his sentence, contending principally that, because there was no proof connecting him to any sale of narcotics in Far Rockaway, (a) the evidence was insufficient to support his conviction on the conspiracy counts, and (b) his sentence was disproportionate to his offense. We see no merit in any of his contentions.
1. Sufficiency of the Evidence as to Narcotics Conspiracy
A defendant may be found guilty of conspiracy on the basis of “evidence from which it can reasonably be inferred that [he] knew of the existence of the scheme alleged in the indictment and knowingly joined and participated in it.” United States v. Gaviria, 740 F.2d 174, 183 (2d Cir.1984). Even a supplier of legal goods to a conspiracy may be found guilty of conspiracy if he supplied those goods “kn[owing] of and intending] to further the illegal venture, ... encouragfing] the illegal use of the goods or halving] a stake in such use.” United States v. Zambrano, 776 F.2d 1091, 1095 (2d Cir.1985); see, e.g., Direct Sales Co. v. United States, 319 U.S. 703, 712-13, 63 S.Ct. 1265, 87 L.Ed. 1674 (1943) (supplier of morphine to physician); United States v. Rush, 666 F.2d 10, 11-12 (2d Cir.1981) (per curiam) (lender of money to marijuana importer). A fortiori one may be found to have been a member of a narcotics conspiracy where he knew the nature of the conspiracy and intended to further it by supplying guns, which we have taken “judicial notice [are], to substantial dealers in narcotics, ... as much tools of the trade as are the commonly recognized articles of narcotics paraphernalia,” United States v. Crespo, 834 F.2d 267, 271 (2d Cir.1987), cert. denied, 485 U.S. 1007, 108 S.Ct. 1471, 99 L.Ed.2d 700 (1988); see, e.g., United States v. Torres, 901 F.2d 205, 235 (2d Cir.), cert. denied, 498 U.S. 906, 111 S.Ct. 273, 112 L.Ed.2d 229 (1990); United States v. Fernandez, 829 F.2d 363, 367 (2d Cir.1987) (per cu-riam).
*201With respect to count one in the present case, there was evidence that Johnson, though he testified that he had never brought guns to New York, in fact drove to New York every week between 1993 and 1995 for the purpose of selling guns. There was evidence that he carried 9-14 guns on each trip, secreting the guns in cubbyholes he created by, for example, removing stereo speakers, and that he sold guns to no fewer than seven members of the present conspiracy. The jury could easily infer that Johnson was aware of the joint venture among many of his buyers. Hamilton, for example, testified that he was introduced to Johnson by Grimes; he was also present when Johnson sold guns to Grimes. And plainly Johnson was aware of the nature of his purchasers' business, for in some instances he was paid not in cash but in narcotics. In sum, the evidence was ample to permit the jury to infer that Johnson knew his purchasers were jointly engaged in narcotics trafficking, that he made numerous and repeated sales to the coconspirators, and that he intended his sales to further their business.
As for count two, which charged conspiracy to distribute near a school zone between September 1996 and April 1998, Johnson's sufficiency challenge, based on the absence of evidence that he sold guns to the coconspirators during that period, is without merit for the reasons stated in Part II.B.1. above. There being no evidence that Johnson withdrew from the conspiracy, in which the jury found he was a participant in finding him guilty on count one, the presumption that his participation continued was sufficient to permit a finding of his guilt on count two.
2. The Claim of Excessive Punishment
Johnson also argues that his sentence to imprisonment for life constitutes cruel and unusual punishment within the meaning of the Eighth Amendment to the Constitution because he was not a partner in the Beach 26th Street organization, did not profit from its business, and did not sell drugs in the New York area. We reject this challenge because, although there was no evidence that Johnson sold drugs in the New York area, the remainder of his factual premise is false.~ The jury found that he in fact was a participant in the Beach 26th Street conspiracy. The evidence was that he sold the coconspirators guns; that he received cash and narcotics in exchange for the guns; and that from his sales of guns and drugs he grossed $30,000 to $40,000 a week, or at least $3 million for 1993-1995. We see no Eighth Amendment violation in his life sentence.
F. Other Contentions
Defendantè also advance other contentions that we find unpersuasive. They include contentions that, the court. erred in discovery rulings, that the prosecutor's summation relied on evidence outside the record, and that the sentences on the drug conspiracy counts were improper.
1. Failure To Disclose Identity of Confidential Informant
During the last six months of the conspiracy, New York City police officer Carl Shepherd participated in numerous undercover narcotics transactions with members of the conspiracy, including four undercover purchases or controlled buys of crack cocaine from Alberto between October 22, 1997 and April 14, 1998. Alberto sought disclosure of the identity of the confidential informant who had introduced Shepherd to Alberto and accompanied the officer on two of his four undercover buys from Alberto. The district court denied his request for disclosure, and Alberto contends that the denial deprived him of a fair *202trial because the informant was the only eyewitness to the events other than the undercover officer.
In order to obtain a new trial on the ground that the government failed to disclose the identity of a confidential informant, a defendant "has the heavy burden of showing that disclosure [wa]s `essential to the defense.'" United States v. Jimenez, 789 F.2d 167, 170 (2d Cir.1986) (quoting Scher v. United States, 305 U.S. 251, 254, 59 S.Ct. 174, 83 L.Ed. 151 (1938)). He must show more than simply that the informant was a participant in or witness to the crime charged, or that the informant might cast doubt on the general credibility of a government witness, see, e.g., United States v. Saa, 859 F.2d 1067, 1073 (2d Cir.1988), cert. denied, 489 U.S. 1089, 109 S.Ct. 1555, 103 L.Ed.2d 858 (1989); "[s]peculation that disclosure of the informant's identity will be of assistance is not sufficient to meet the defendant's burden," United States v. Fields, 113 F.3d 313, 324 (2d Cir.), cert. denied, 522 U.S. 976, 118 S.Ct. 434, 139 L.Ed.2d 334 (1997). We will not disturb the district court's disclosure rulings in the absence of an abuse of discretion. See, e.g., DiBlasio v. City of New York, 102 F.3d 654, 657 (2d Cir.1996).
Here, Aiberto offered only his conjecture that the informant's testimony would be significant, and it seems plain that any help from the informant would instead have been minimal. For example, as the district court noted, the informant witnessed only two of the four meetings between the undercover officer and Alberto; the government introduced crack cocaine purchased from Alberto at a meeting unattended by the informant. In addition, the government's evidence included drug-trafficking paraphernalia seized from Alberto's home pursuant to a search warrant, as well as Alberto's postarrest statements in May 1998 that he had been selling crack cocaine on Beach 26th Street for some two-to-three years, that he received the crack from those who controlled the block, that he sold it himself or through his own agents, that he turned the proceeds over to those who controlled the block, and that these activities earned him more than $150 a day. We see no abuse of discretion in the district court's ruling that Alberto had not met his burden of establishing how the informant's identity would be helpful to his defense.
2. The Provenance of G'rimes's Weapons
Grimes contends that he is entitled to a new trial because the government's attorney stated in summation that a certain gun possessed by Grimes was made in Italy and that others he possessed were made in California, and that those facts were not in the record. In order to prevail on such a claim, Grimes must demonstrate "(1) that the prosecutor's remarks were improper and (2) that the remarks, taken in the context of the entire trial, resulted in substantial prejudice," United States v. Bautista, 23 F.3d 726, 732 (2d Cir.), cert. denied, 513 U.S. 862, 115 S.Ct. 174, 130 L.Ed.2d 110 (1994), or "`"so infected the trial with unfairness as to make the resulting conviction a denial of due process,"'" id. at 734 (quoting Darden v. Wainwright, 477 U.S. 168, 181, 106 S.Ct. 2464, 91 L.Ed.2d 144 (1986) (quoting Donnelly v. DeChristoforo, 416 U.S. 637, 643, 94 S.Ct. 1868, 40 L.Ed.2d 431 (1974))). Grimes has not met this standard.
Assuming that the prosecutor's references to California and Italy were improper, we cannot see that those references were fundamentally unfair or caused Grimes any substantial prejudice. The government was not required to prove where the guns were manufactured but only that they had traveled in or affected *203interstate or foreign commerce, see 18 U.S.C. § 922(g)(1) (unlawful for prior convicted felon to “possess” a firearm “in or affecting commerce ... or to receive any firearm ... which has been shipped or transported in interstate or foreign commerce”). Given the testimony of Hamilton that Grimes bought guns from Johnson in New York, and the testimony of Hale that the guns Johnson sold in New York had been transported from the Virginia-Tennessee area, it was plainly inferable that the guns Grimes possessed had traveled in and affected interstate commerce. Accordingly, any error of the prosecutor in referring to specific points of origin for which there was no evidence in the record was entirely harmless.
3. Sentences Premised on Cocaine Base Rather Than Marijuana
Defendants also contend that because they were convicted of conspiracy to distribute and possess cocaine base and marijuana, they are entitled to have their sentences calculated on the basis of the controlled substance that carries the more lenient statutorily prescribed sentence, ie., marijuana. We reject this contention.
Ordinarily if the jury returns a general verdict of guilty on a conspiracy count involving more than one controlled substance and does not indicate on which controlled substance it based its verdict, we will assume that the conviction was for the substance that carries the most lenient statutorily prescribed sentence for which the evidence was sufficient to support a conviction. See United States v. Barnes, 158 F.3d 662, 668 (2d Cir.1998); United States v. Orozco-Prada, 732 F.2d 1076, 1083 (2d Cir.), cert. denied, 469 U.S. 845, 105 S.Ct. 154, 83 L.Ed.2d 92 (1984). In the present case, however, the jury’s verdict of guilty against each defendant on counts one and two was accompanied by a finding that the government proved beyond a reasonable doubt that each that defendant conspired to distribute and to possess with intent to distribute crack cocaine. Thus, there is no possibility that the jury might have found any defendant guilty of conspiring to deal only in marijuana. It was appropriate for the district court to sentence defendants in accordance with the penalties for trafficking in crack cocaine rather than those for marijuana.
G. The Count-One Convictions of Bruce, Carf, and Grimes
Finally, as indicated at the outset of this opinion, the government concedes the correctness of the contention of Bruce, Carf, and Grimes that under Rutledge v. United States, 517 U.S. 292, 116 S.Ct. 1241, 134 L.Ed.2d 419, they could not be convicted of both continuing criminal enterprise and a narcotics conspiracy that is a lesser included offense of the CCE. This principle requires the reversal of the convictions of those three defendants on count one of the superseding indictment.
We note that the Rutledge principle does not affect the convictions of these defendants on count two of the superseding indictment, which charged an unlawful conspiracy to distribute narcotics in proximity to a school zone in violation of 21 U.S.C. § 860. Because one element of the § 860 offense, ie., proximity to a school zone, is not an element in the CCE offense, the § 860 offense is not included in the CCE offense.
CONCLUSION
We have considered all of defendants’ contentions on these appeals and, except as indicated above, have found in them no basis for reversal. The convictions of Bruce, Carf, and Grimes on count one are reversed, and the matter is remanded to the district court for entry of amended *204judgments as to those defendants, dismissing count one. In all other respects, the judgments of conviction are affirmed.