Jayme Walker met W.F., who was fourteen at the time, "on a social media site for guys seeking other people." W.F. falsely claimed that he was eighteen years old to join the site, but he told Walker that he was fourteen. The two exchanged sexual messages and images on the website and by text message. The communications began in June 2013 and ended in February 2014 when W.F.'s mother discovered the exchanges and a police investigation began.
Walker waived indictment and was charged by information with four counts. The Government later dismissed one count. Following trial, a jury found him guilty on the remaining three counts: (1) 18 U.S.C. § 1470, Transfer of Obscene Materials to a Minor; (2) 18 U.S.C. § 2252(a)(2) and (b)(1), Receipt of Child Pornography; and (3) 18 U.S.C. § 2251(a), Sexual Exploitation of a Minor. The district court1 sentenced Walker to 264 months' imprisonment.
*1008Walker now appeals. He claims that the district court erred by refusing to admit evidence of W.F.'s sexual conversations with other men. He argues further that the district court erred in excluding evidence about the source of Walker's fantasies and in limiting his cross-examination of the victim. He also contends the district court abused its discretion by failing to require knowledge of the victim's age in a jury instruction and by denying his motion for judgment of acquittal. Finally, he argues that his sentence constitutes cruel and unusual punishment in violation of the Eighth Amendment. We consider each of these arguments in turn and affirm.
I.
A.
"We review a district court's interpretation and application of the rules of evidence de novo and its evidentiary rulings for abuse of discretion." United States v. Street , 531 F.3d 703, 708 (8th Cir. 2008). "However, we review evidentiary rulings de novo when they implicate constitutional rights." United States v. Pumpkin Seed , 572 F.3d 552, 558 (8th Cir. 2009).
First, Walker argues that the district court erred when it excluded evidence of W.F.'s sexual communications with other men. In a case involving "alleged sexual misconduct," the Federal Rules of Evidence prohibit the admission of evidence "offered to prove that a victim engaged in other sexual behavior" or "evidence offered to prove a victim's sexual predisposition." Fed. R. Evid. 412(a)(1), (2). But the rule contains three exceptions for criminal cases. Fed. R. Evid. 412(b). Only the first and third exceptions are at issue here. The first exception allows "evidence of specific instances of a victim's sexual behavior, if offered to prove that someone other than the defendant was the source of semen, injury, or other physical evidence." Fed. R. Evid. 412(b)(1)(A). The third exception allows "evidence whose exclusion would violate defendant's constitutional rights." Fed. R. Evid. 412(b)(1)(C).
Walker argues that the first exception applies in this case "because the images sent by [W.F. to Walker] could have been of himself or other people, or both." And he says the images "could have been prepared by [W.F.] for another person or during another conversation with someone else," rather than at Walker's inducement. Walker sought to "adduce evidence that [W.F.] initiated communication with other adults through age-restricted websites; and sent nude photographs using age-restricted websites, during the same time that he is alleged to have communicated with Defendant." The district court questioned the relevance of the communications and observed that Walker appeared to be arguing that "exploitation by this defendant isn't as bad because lots of folks were exploiting [W.F.]." It then excluded the communications.
We find no basis for concluding that the district court abused its discretion in excluding the evidence. The first exception to Rule 412 allows "evidence of specific instances of a victim's sexual behavior, if offered to prove that someone other than the defendant was the source of ... other physical evidence." But here, Walker sought to question W.F. about the general fact that he had initiated sexual conversations with and sent images of himself to other people. He did not seek to introduce "specific instances" that indicated that others had induced W.F. to produce the images that W.F. later sent to Walker. Instead, Walker argued that the evidence he sought to introduce showed that W.F. "was initiating with everybody. This is what he did. This was his hobby with everybody."
*1009Thus, it was not an abuse of discretion for the district court to exclude the evidence. See United States v. Ogden , 685 F.3d 600, 604-05 (6th Cir. 2012) (excluding evidence of the victim's chat logs with other men offered by the defendant to show that "one of those men might have originally persuaded the victim to take the explicit pictures" in a § 2251 case because Rule 412(a)(1)"forbids the introduction of 'evidence offered to prove that a victim engaged in other sexual behavior' ").
Walker also argues that the third Rule 412 exception applies in this case because evidence that W.F. "sent and received numerous sexual communications with a number of people during the same period of time he sexted Appellant directly contradicts" an element of Count Three: that Walker used, persuaded, induced, or enticed W.F. to produce the image. 18 U.S.C. § 2251(a). Walker claims that the exclusion of the evidence thus violated his Fifth and Sixth Amendment rights to "introduce evidence in his own defense."
When considering whether evidence is admissible under Rule 412(b)(1)(C), "we start with the premise that defendants have a constitutional right under the Fifth and Sixth Amendments to introduce evidence in their defense." Pumpkin Seed , 572 F.3d at 559. But the right "is not without limitation." Id . at 560. "[T]he key inquiry ... is whether the district court's exclusion of evidence ... was arbitrary or disproportionate to the purposes that its exclusion was designed to serve." Id . Here, the district court "saved [W.F.] from the harassment and embarrassment concomitant with discussing the details" of his sexual conversations with other men. See id . And, as explained above, Walker sought to introduce evidence of W.F.'s sexual history, rather than "specific instances" indicating that others had induced W.F. to produce the images that W.F. sent to Walker. Thus, the exclusion of the evidence was not "arbitrary or disproportionate to the purposes that its exclusion was designed to serve." Id . The district court did not violate Walker's Fifth and Sixth Amendment rights by excluding the evidence of W.F.'s sexual communications with other men.2
Second, Walker argues that the district court erred in excluding "fantasy source material." Walker maintained at trial that he believed that W.F. was an adult and that he and W.F. were acting out a fantasy. In support of that argument, he sought to admit an exhibit containing pages from a website from which Walker found "source material" for fantasies that he played out in conversations on social media sites like the one on which he met W.F. The district court determined that the exhibit was not helpful or probative and that it would be confusing to the jury. The exhibit is a twelve-page list of categories into which the website grouped fantasy stories. Although it was not admitted into evidence, Walker was permitted to testify about this information. He explained that the website he used contained "categories" of information, including "athletics," "bestiality," "celebrity," and a "cross-generational" category that "deals with older guys and younger guys." Walker also testified that the website is "a huge *1010listing of any possible scenario that you can think of in the erotica world." Thus, any error is harmless because the exhibit was cumulative. See McWilliams v. United States , 394 F.2d 41, 45-46 (8th Cir. 1968) ("[I]n view of the cumulative nature of the rejected evidence, its denial can hardly be said to have prejudiced the appellant."); United States v. Tapio , 634 F.2d 1092, 1094-95 (8th Cir. 1980) (per curiam) ("The court's error, if any, in excluding the evidence was harmless. ... The excluded testimony was merely cumulative.").
B.
Walker next argues that the district court erred in placing a time limit on his cross-examination of W.F. The district court limited Walker's cross-examination to one hour and thirty minutes, four times the length of the Government's direct examination. In the past, we have applied competing standards of review. Compare United States v. Warfield , 97 F.3d 1014, 1024 (8th Cir. 1996) ("Absent a clear abuse of discretion and a showing of prejudice, we will not reverse a district court's ruling limiting cross-examination of a prosecution witness on the basis that it impermissibly infringed upon the defendant's right of confrontation.") with United States v. Plume , 847 F.3d 624, 629 (8th Cir. 2017) ("This court reviews evidentiary rulings regarding the scope of a cross examination for abuse of discretion, but where the Confrontation Clause is implicated, we consider the matter de novo." (internal quotation marks omitted)). The outcome is the same under either standard.
"A critical factor in determining whether a defendant's right of confrontation has been violated is whether the defendant had other ways to obtain the effect that the excluded examination would have allegedly established." United States v. Brown , 110 F.3d 605, 611 (8th Cir. 1997). The district court correctly noted that most of the cross-examination had been spent cumulatively reading an exhibit containing the texts between W.F. and Walker that had been admitted into evidence and would be available to the jury. And it concluded that it was "not efficient to go through a lengthy 264-page document." Further, Walker presents no evidence that the time limit prejudiced him. See Harrington v. Iowa , 109 F.3d 1275, 1277 (8th Cir. 1997) (requiring a defendant to show that "[a] reasonable jury might have received a significantly different impression of [the witness'] credibility had [defense] counsel been permitted to pursue his proposed line of cross examination" (alteration in original)). Thus, we will not reverse the district court's ruling.
C.
In addition, Walker argues that the district court abused its discretion by not instructing the jury that Walker had to know that W.F. was a minor as he claims is required for a conviction under 18 U.S.C. § 2251(a). "We review a district court's formulation of jury instructions for abuse of discretion and consider whether the instructions correctly state the applicable law." United States v. Pliego , 578 F.3d 938, 942 (8th Cir. 2009) (internal quotation marks omitted). We have previously held that a "district court did not abuse its discretion in refusing to instruct the jury that knowledge of the victim's age is an element of § 2251(a)." Id. at 943. Thus, the district court did not abuse its discretion here.3
*1011D.
Walker further argues that the district court erred in denying his motion for judgment of acquittal. He contends that there was insufficient evidence to support his conviction because "all evidence of guilt originated from a platform of fantasy." Specifically, Walker claimed in his motion that all of the photographs he received from W.F. were of a "post-pubescent male" and that it was "impossible" to know, based on those images, that W.F. was less than the required age.4 He argued that "there was no link between Defendant and the reality that the victim was in fact a minor, other than the statements made ... that from Defendant's perspective were fantastical."
"We review the sufficiency of the evidence de novo, viewing evidence in the light most favorable to the government, resolving conflicts in the government's favor, and accepting all reasonable inferences that support the verdict." United States v. Piwowar , 492 F.3d 953, 955 (8th Cir. 2007). "We must uphold the verdict if there is an interpretation of the evidence that would allow a reasonable-minded jury to find the defendant guilty beyond a reasonable doubt." United States v. Wainright , 351 F.3d 816, 822 (8th Cir. 2003).
Walker communicated with W.F. for approximately eight months. W.F. testified that he told Walker that he was fourteen, and he testified that he did not think his communications with Walker were based on fantasy. W.F. also sent Walker a message about working on homework and another about attending the homecoming dance. Further, Walker's own messages support the jury's finding. On one occasion Walker said, "I can't lie. It kind of makes me nervous. I'm not one of those pedophile guys, but you are underage and can get me in trouble." On a different occasion, after W.F. asked Walker to send him photographs, Walker replied, "I don't think we should that [sic] anymore because of our age difference. I can get in trouble. Sorry." And there is no dispute that Walker and W.F. sent each other pornographic images. This evidence is sufficient for a reasonable jury to conclude that Walker knew that W.F. was less than sixteen years old. See also Lavender v. Kurn , 327 U.S. 645, 653, 66 S.Ct. 740, 90 L.Ed. 916 (1946) ("[W]here, as here, there is an evidentiary basis for the jury's verdict, the jury is free to discard or disbelieve whatever facts are inconsistent with its conclusion."). Thus, the district court did not err in denying Walker's motion for judgment of acquittal.
E.
Finally, Walker argues that his 264-month sentence constitutes cruel and unusual punishment in violation of the Eighth Amendment. We review Eighth Amendment challenges de novo . United States v. Vanhorn , 740 F.3d 1166, 1169 (8th Cir. 2014). The Eighth Amendment does not guarantee proportionality between crime and sentence. See Harmelin v. Michigan , 501 U.S. 957, 965, 111 S.Ct. 2680, 115 L.Ed.2d 836 (1991). "[T]his Court has never held that a sentence within the statutory range violates the Eighth Amendment." United States v. Patten , 664 F.3d 247, 252 (8th Cir. 2011) (alteration in original). Walker's sentence is less than the 360-month statutory maximum for Walker's § 2251 conviction. And we have *1012previously rejected an Eighth Amendment challenge to a 360-month sentence under § 2251. See United States v. Martynenko , 717 F. App'x 645, 646 (8th Cir. 2018) (per curiam) (rejecting an Eighth Amendment challenge to a sentencing package that included a 360-month sentence for a § 2251(a) conviction); Patten , 664 F.3d at 252. Likewise, we reject Walker's claim that his 264-month sentence constitutes cruel and unusual punishment in violation of the Eighth Amendment.
II.
For the foregoing reasons, we affirm Walker's conviction and sentence.