Derwin Ard appeals his 24-month sentence on two grounds: the district court incorrectly calculated his guidelines range, and it relied on irrelevant factors in sentencing him. We affirm.

I.

On March 1, 2010, Ard went to Fleet Safety Equipment, located in Memphis, where he told staff members he worked for the United States Department of Justice and he needed a law enforcement lighting package installed on his vehicle. Unable to help him that day, Fleet Safety told him to come back later. Two days later, Ard returned and again told company employees he worked for the Department of Justice and again asked for a lighting installation. Fleet Safety installed the lighting package the next day.

When it came time to pay for the installation (around $2,500), Ard’s credit card was declined. A Fleet Safety employee permitted him to pay with a check on the assumption that, as a law enforcement officer, he could be trusted. The check turned out to be worthless. Fleet Safety contacted the police, who arrested Ard in Little Rock, Arkansas a few days later. As it turns out, Ard was not an employee of the Department of Justice, he was driving a stolen vehicle and he had a lengthy criminal history, including convictions for impersonation-related offenses in four States.

A grand jury indicted Ard for impersonating a federal officer. See 18 U.S.C. § 912. He pled guilty and received a 24-month sentence, nine months above the guidelines range of nine to fifteen months.

II.

Ard challenges the procedural reasonableness of his sentence, claiming the district court improperly calculated his guidelines range. In calculating Ard’s guidelines range, the district court started with the impersonation guideline, U.S.S.G. § 2J1.4, which initially calls for an offense level of six. The guideline adds the following adjustment: “If the impersonation *532 was to facilitate another offense, apply the guideline for an attempt to commit that offense, if the resulting offense level is greater than the offense level determined [by this guideline].” U.S.S.G. § 2J1.4(c)(l). Because the impersonation facilitated a fraud valued at more than $5,000 (as no one disputes), the district court applied the § 2J 1.4(c)(1) cross-reference and sentenced Ard under the higher offense level (eight) required for frauds of this value. See U.S.S.G. § 2B1.1.

Ard protests that the district court should not have stopped there. It should have applied the cross-reference in the general fraud guideline, which refers back to the impersonation guideline. U.S.S.G. § 2Bl.l(c)(3). That cross-reference applies where “the defendant was convicted under a statute proscribing false, fictitious, or fraudulent statements or representations generally (e.g., 18 U.S.C. § 1001, § 1341, § 1342, or § 1343)” and where the conduct set forth in the offense is “specifically covered by another guideline.” U.S.S.G. § 2B1.1(c)(3). As Ard sees it, this interpretation creates an “infinite loop” in the sentencing guidelines, as § 2J1.4 refers to § 2B1.1, which refers back to § 2J1.4 and so on-begging the question whether the original offense level (six) or the cross-referenced offense level (eight) applies. Any doubt about which offense level is the right one, Ard adds, implicates the rule of lenity and entitles him to the more lenient interpretation.

Not quite. Even if these guidelines do create a loop, it is not infinite. The impersonation guideline says to apply the “greater ... offense level” — which is to say eight, not six — when a defendant’s impersonation facilitates another offense. U.S.S.G. § 2J1.4(c)(l). Whether one starts with the fraud guideline or the impersonation guideline or ends with either guideline, all of these possibilities lead to an offense level of eight for a fraud caused by impersonation valued at over $5,000. Any loop ends up in the same place: an offense level of eight.

The cross-reference found in the general fraud guideline (§ 2Bl.l(c)(3)), at any rate, does not apply to him. It applies to defendants convicted under statutes prohibiting “false, fictitious, or fraudulent statements or representations generally.” U.S.S.G. § 2Bl.l(c)(3). Yet Ard was convicted under a specific statute prohibiting the impersonation of a federal employee. 18 U.S.C. § 912. The statutes listed as examples in § 2Bl.l(c)(3) confirm the distinction. They generally prohibit fraudulent statements to government officials, 18 U.S.C. § 1001, general mail fraud, id. § 1341, mail fraud through the use of fictitious names or addresses, id. § 1342, and general wire fraud, id. § 1343. None of these statutes covers this offense.

Ard also challenges the substantive reasonableness of his sentence, claiming the district court relied too heavily on the fact that Ard received a military discharge under “other than honorable” conditions. Early in the sentencing hearing, the court asked Ard about his military record and spent some time sorting through it because Ard’s account and the official records did not line up. When it came time to explain its upward variance, however, the court did not mention the military discharge in discussing the § 3553(a) factors. Rather, it focused on the need to protect the public from Ard’s crimes and to deter Ard from committing future crimes given his history of serial impersonations. 18 U.S.C. § 3553(a)(2); see United States v. Tate, 516 F.3d 459, 471 (6th Cir. 2008). Ard’s record of military service simply did not come up during this phase of the hearing. Given the need to protect the public from a repeat impersonator, the *533 court’s 24-month sentence was a reasonable one.

III.

For these reasons, we affirm.