Susan Frank and Jane Frank Kresch appeal from judgments entered by the United States District Court for the Eastern District of New York (Jack B. Weinstein, Judge) convicting them, after a jury trial, of conspiracy to commit mail fraud, in violation of 18 U.S.C. §§ 371 and 1341. ^[1] The district court sentenced Frank principally to three years of probation, including eight months of home confinement, and a $10,000 fine. It sentenced Kresch principally to a term of imprisonment of one year and one day, two years of supervised release, and a $10,000 fine. The appellants’ principal claims on appeal are: (1) that there was insufficient evidence of an intent to commit mail fraud; (2) that the district court erred in its instruction on “intent to defraud;” (3) that the district court’s overt act instruction constructively amended the indictment; and (4) that the district court’s statute of limitations instruction impermissibly allowed the jury to convict the appellants on the basis of a potentially time-barred conspiracy.

We affirm.

I.

We describe below the evidence before the jury, which we are required to view in the light most favorable to the government. See Glasser v. United States, 315 U.S. 60, 80, 62 S.Ct. 457, 86 L.Ed. 680 (1942); United States v. LaBarbara, 129 F.3d 81, 82 (2d Cir.1997). Frank and Kresch were the comptroller and “sludge coordinator,” ^[2] respectively, of General Marine Transport Corporation (“General Marine”). For many years, including the period from January 1988 through January 1989, General Marine contracted with various municipalities in New York and New Jersey to transport raw sewage sludge by barge for disposal at sea. In keeping with federal regulations that were phased in during 1987 and took effect took effect in 1988, the con[*335] tracts called for General Marine to dispose of the waste at a site 106 miles from shore. Prior to 1987, federal law had required that sewage sludge be disposed of only 12 miles from shore. As a result of the new legal requirement, General Marine’s municipal customers paid two to four times more for disposal at the 106-mile site.

Throughout 1988, General Marine’s barges routinely dumped sludge in waters well short of the 106-mile site. Nonetheless, General Marine billed its municipal customers—by mail—-as if the sludge were being dumped in accordance with General Marine’s contractual and legal obligations. Both Frank and Kreseh were actively involved in General Marine’s billing practices. Moreover, for the purpose of concealing General Marine’s “short-dumping,” both defendants participated in the falsification of billing-related records.

Based on the practices described above, an indictment returned on July 1, 1993 charged the appellants and others with conspiracy to defraud various public entities in New York and New Jersey, in violation of 18 U.S.C. §§ 371 and 1341. The indictment also charged the appellants and others with several environmental crimes. Judge Edward Korman, to whom the case was initially assigned, severed the environmental crime counts and tried those counts first; the appellants were acquitted. The appellants were then tried before Judge Weinstein on the single count of conspiracy to commit mail fraud and were convicted. ^[3] This appeal followed.

II.

A. Sufficiency of the Evidence

The appellants’ first claim is that there was insufficient evidence to support their convictions for conspiracy to commit mail fraud because the government purportedly failed to prove that the appellants had any intent to harm their municipal customers. See United States v. D’Amato, 39 F.3d 1249, 1257 (2d Cir.1994) (18 U.S.C. § 1341 requires that the government prove that the defendant contemplated some actual harm or injury to the victim). The appellants maintain that the municipalities got precisely what they bargained for—sludge disposal— and that, accordingly, the appellants neither intended nor caused the municipalities any harm. The appellants premise their argument on the assertion that it was irrelevant to the municipalities that not all of the sludge was dumped at the 106-mile site. This challenge to the sufficiency of the evidence “faces a heavy burden, because we must view the evidence in the light most favorable to the government and ask only whether a rational jury could find beyond a reasonable doubt” that the appellants intended or contemplated some harm to the municipalities. LaBar-bara, 129 F.3d at 84. We believe that the appellants have failed to meet that heavy burden here and that there was ample evidence to support the appellants’ convictions.

A defendant who has used the mails fraudulently to bill a customer for services that have not been provided may not defeat a mail fraud charge simply by providing alternative services. See United States v. Wallach, 935 F.2d 445, 461 (2d Cir.1991); see also United States v. Paccione, 949 F.2d 1183, 1196 (2d Cir.1991) (“Use of the mails in furtherance of a scheme to offer services in exchange for a fee, with the intent not to perform those services, is within the reach of § 1341.”), cert. denied, 505 U.S. 1220, 112 S.Ct. 3029, 120 L.Ed.2d 900 (1994). The evidence in this case showed that the municipalities contracted for disposal at the 106-mile site, that they paid a premium for that service, that General Marine in fact routinely dumped sludge in waters short of the 106-mile site, and that the appellants nonetheless participated in a fraudulent scheme to bill the municipalities by mail as if all of the waste disposal had occurred at the 106-mile site. Moreover, a representative of one of the municipalities testified that General Marine’s short-dumping could have subjected the municipality to fines and to the loss of its environmental permit. This evidence suffices to establish that the appellants intended to harm the municipalities.

[*336] The appellants’ reliance on United States v. Starr, 816 F.2d 94 (2d Cir.1987), is misplaced. The defendants in that ease were operators of a bulk mailing service who cheated the Postal Service by concealing higher-rate packages in mail sacks that were supposed to contain only lower-rate packages. See id. at 96. Although the Postal Service was victimized by the scheme, the indictment charged the defendants only with defrauding their own customers. See id. at 97 n. 4. We reversed the defendants’ mail fraud convictions, on the ground that the customers had received the same service they would have received had the defendants paid the proper postage. See id. at 99. By contrast, in this case the municipalities did not receive the service—sludge disposal at the 106-mile site—for which they had contracted and paid. Accordingly, Starr is of no help to the appellants here.

B. Jury Instruction on “Intent to Defraud”

The appellants next claim that the district court’s instruction on “intent to defraud” was in error. The district court instructed the jury as follows:

Intent to defraud means to act knowingly and with the specific intention to deceive for the purpose of causing some financial loss to another or gain to oneself or to another person.

(emphasis added). The appellants argue that this instruction impermissibly allowed the jury to convict on the basis that the appellants intended to cause either loss to their customers or gain to themselves, even though only the former satisfies the mail fraud statute’s element of fraudulent intent. In support of their argument, the appellants again rely on Starr. There, we found that, “[t]o the extent that [a virtually identical] charge permit[ted] the jury to find an intent to defraud based solely on the defendants’ appropriation of a benefit to themselves, it [was] error.” Starr, 816 F.2d at 101. We believe that the appellants’ claim in the instant case is without merit.

In light of the appellants’ conceded failure to object at trial, we review the district court’s instruction for plain error, see Fed.R.Crim.P. 52(b) (plain error standard)— a standard that requires “ ‘(1) error, (2) that is plain, and (3) that affect[s] substantial rights. If all three conditions are met, an appellate court may then exercise its discretion to notice a forfeited error, but only if (4) the error seriously affect[s] the fairness, integrity, or public reputation of judicial proceedings.’” United States v. Rossomando, 144 F.3d 197, 200 (2d Cir.1998) (quoting Johnson v. United States, 520 U.S. 461,-, 117 S.Ct. 1544, 1549, 137 L.Ed.2d 718 (1997)). To determine if there was plain error, we will review an “erroneous instruction in light of other instructions and the evidence to determine what findings the jury made and whether such findings are the functional equivalent of required findings.” United States v. Chandler, 98 F.3d 711, 715 (2d Cir.1996) (internal quotation marks omitted).

In Chandler, we reviewed for plain error an instruction in a bank fraud case under 18 U.S.C. § 1344, which we recognized to have been modeled on the mail and wire fraud statutes. See id. at 715. The defendant in Chandler had used false identification to obtain a line of credit from a bank and then made purchases using that line of credit. See id. at 713. The trial judge failed to instruct the jury that, in order to convict the defendant of bank fraud, it had to find an intent to cause the bank actual harm; instead, the judge instructed the jury only that it needed to find an intent to deceive. We found that the instruction was in error, because an intent to harm is an essential element of bank fraud. See id. at 714-15. Nonetheless, we observed that, “[e]ven though the court did not require a finding of intent to harm in its instructions defining ‘scheme to defraud,’ this error is not prejudicial if the jury’s finding of intent to deceive is the ‘functional equivalent’ of a finding of intent to harm.” Id. (internal quotation marks omitted). We reasoned that Chandler’s deceit in obtaining the line of credit was “inexplicable other than as a means of intentionally exposing [the bank] to an unwanted risk.” Id. at 716. Moreover, because “the intent to harm [was] inextricably bound to the intent to deceive” we concluded that the jury’s finding of intent to deceive was the “functional equivalent of the essen[*337] tial finding of intent to harm.” Id. Accordingly, we found that the defendant had not been prejudiced by the instruction and that there was no plain error.

Similarly here, it was error for the district court to instruct the jury that it could find an intent to defraud based solely on the appellants’ desire to gain a benefit for themselves. See Starr, 816 F.2d at 101. However, as in Chandler, we believe that this error worked no prejudice. The appellants’ intended gain to themselves was inextricably bound to a corresponding loss to their customers. The essence of the appellants’ scheme was to obtain money by inducing their customers to pay for services that were not provided. Because the jury here presumably found that the appellants intended to gain from their scheme, it follows that it also found the “functional equivalent”—an intended loss to the appellants’ customers. Accordingly, the appellants were not prejudiced by the jury instruction and there was no plain error.

C. Jury Instruction on the Overt Act Requirement

The appellants next claim that the district court’s jury instruction regarding the overt act element of the § 371 conspiracy charge constructively amended the indictment.

The district court’s instruction, in pertinent part, was as follows:

The third element of the crime that the government has to establish beyond a reasonable doubt is that at least one of the overt acts charged in the indictment or one which is substantially the same as one explicitly charged was knowingly committed by at least one of the conspirators for the purpose of carrying out the unlawful agreement. ... You have to agree on at least one overt act charged or substantially the same as one explicitly charged.

(emphasis added).

The appellants assert that the district court’s use of “substantially the same” constructively amended the indictment, in violation of the grand jury clause of the Fifth Amendment. ^[4] The argument is without merit.

To prevail on a constructive amendment claim, a defendant must demonstrate that either the proof at trial or the trial court’s jury instructions so altered an essential element of the charge that, upon review, it is uncertain whether the defendant was convicted of conduct that was the subject of the grand jury’s indictment. See United States v. Zingaro, 858 F.2d 94, 98 (2d Cir.1988). In the case before us, the appellants do not claim that the district court permitted the government to introduce any evidence relating to conduct that differed significantly from that described in the indictment. Nor do the appellants claim that the district court’s instruction referred to any such evidence. In the absence of any indication that the evidence adduced at trial impermissibly broadened the basis of the appellants’ convictions, we cannot accept the argument that the district court’s mere use of the “substantially the same” language constructively amended the indictment.

In fact, the district court’s instruction is entirely consistent with the well-established rule of this and other circuits that the overt act element of a conspiracy charge may be satisfied by an overt act that is not specified in the indictment, at least so long there is no prejudice to the defendant. ^[5] See, e.g., [*338] United States v. Bryan, 122 F.3d 90, 93 (2d Cir.1997); United States v. Fassoulis, 445 F.2d 13, 19 (2d Cir.1971); United States v. Armone, 363 F.2d 385, 400-01 (2d Cir.), cert. denied, 385 U.S. 957, 87 S.Ct. 391, 392, 398, 17 L.Ed.2d 303 (1966); United States v. Negro, 164 F.2d 168, 173 (2d Cir.1947); see also United States v. Schurr, 794 F.2d 903, 907 n. 4 (3d Cir.1986); United States v. Morales, 677 F.2d 1, 2 (1st Cir.1982); United States v. Clay, 495 F.2d 700, 706 (7th Cir.), cert. denied, 419 U.S. 937, 95 S.Ct. 207, 42 L.Ed.2d 164 (1974); Brulay v. United States, 383 F.2d 345, 350-51 (9th Cir.), cert. denied, 389 U.S. 986, 88 S.Ct. 469, 19 L.Ed.2d 478 (1967); Strauss v. United States, 311 F.2d 926, 932 (5th Cir.), cert. denied, 373 U.S. 910, 83 S.Ct. 1299, 10 L.Ed.2d 412 (1963). These cases reflect the fact that, “[b]ecause proof at trial need not, indeed cannot, be a precise replica of the charges contained in the indictment, this court has consistently permitted significant flexibility in proof, provided that the defendant was given notice of the core of criminality to be proven at trial.” United States v. Heimann, 705 F.2d 662, 666 (2d Cir.1983) (internal quotation marks omitted); accord United States v. Patino, 962 F.2d 263, 266 (2d Cir.1992). There is no indication in the record here that the evidence adduced at trial unfairly surprised the appellants or prejudiced them in any other way. If anything, the district court’s use of the “substantially the same” language avoided prejudice to the defendants by ensuring that they would not be convicted on the basis of an overt act that differed in any significant way from the overt acts specified in the indictment.

D. Jury Instruction Regarding the Statute of Limitations

The appellants’ final contention, closely related to the one we have just considered, is that the district court’s instruction on the statute of Imitations allowed the jury to convict on the basis of a potentially time-barred conspiracy. Again, the appellants’ argument is that the district court improperly allowed the jury to find the statute of limitations satisfied by an act “substantially the same” as one charged in the indictment. The relevant instruction, to which the appellants objected before the district court, ^[6] is as follows:

The fourth element the government must prove is that this prosecution is not barred or precluded by what is known as the statute of limitations....

So in this case the government must prove that the conspiracy agreement charged was in existence on July 2, 1988 and that at least one overt act in furtherance of the conspiracy was performed after that date. Accordingly you must return a verdict of not guilty unless you find that the government has proven beyond a reasonable doubt that the conspiracy charged in the indictment continued on or after July 2, 1988 and that an overt act was performed after that date as I’ve defined overt act for you. 7

(emphasis added)

The appellants argue that, even if the government may satisfy the overt act element of § 371 by proof of an act not explicitly[*339] charged in the indictment, the statute of limitations element may only be satisfied by an act that is explicitly charged. According to the appellants, this is an issue of first impression in this Circuit.

We see no reason to devise a special rule for the statute of limitations requirement that is different from the rule governing the overt act element. In the matter of the statute of limitations, as in the case of overt acts generally, our principal concerns are that a defendant have fair and adequate notice of the charges against him and have suffered no undue prejudice as a result of the proof offered. See Schurr, 794 F.2d at 907 n. 4 (“It is well settled that the government can prove overt acts not listed in the indictment, so long as there is no prejudice to the defendants thereby. There would appear to be no reason that the government could not satisfy its requisite showing under the statute of limitations by means of an overt act not listed in the indictment.”) (citations omitted). In other words, the statute of limitations may be satisfied by proof of an overt act not explicitly listed in the indictment, as long as a defendant has had fair and adequate notice of the charge for which he is being tried, and he is not unduly prejudiced by the asserted variance in the proof.

In this ease, there is no indication that the appellants were prejudiced by the district court’s instruction on the statute of limitations requirement. The grand jury returned the indictment on July 1, 1993. Accordingly, pursuant to the five-year limitations period set forth in 18 U.S.C. § 3282, the government was required to prove that at least one overt act in furtherance of the conspiracy was committed after July 1,1988. ^[8] The indictment alleged at least two timely overt acts and the government offered evidence to support both: in August 1988, the appellants caused a General Marine employee to falsify records to conceal illegal dumping; and on October 30, 1988, a General Marine barge illegally dumped sludge near New Jersey, short of the 106-mñe site required by law and by contract. ^[9] By instructing the jury both (1) that it had to find that a timely overt act was committed and (2) that the timely overt act had to be either listed in the indictment or substantially the same as an act listed in the indictment, the district court scrupulously avoided any prejudice to the appellants. ^[10]

III.

To summarize:

(1) The appellants’ challenge to the sufficiency of the evidence fails because there was ample proof that the appellants intended to harm General Marine’s municipal customers by billing them for services that General Marine did not provide.

(2) The appellants’ challenge to the district court’s instruction on “intent to defraud” fails because, under the circumstances presented here, the jury’s finding of an intended gain to the appellants was the “functional equivalent” of an intended harm to General Marine’s municipal customers.

(3) The appellants’ challenge to the district court’s overt act instruction fails because the appellants did not demonstrate that the instruction constructively amended the indict[*340] ment by impermissibly broadening the basis for their convictions.

(4) The appellants’ challenge to the statute of limitations instruction fails because we reject their contention that the statute of limitations may only be satisfied by an act listed in the indictment and because the appellants were not prejudiced by the district court’s instruction.

We have carefully considered all of the appellants’ arguments and find them to be without merit. ^[11]

The judgments of the district court are affirmed.