
628 F.3d 923 (2010)
UNITED STATES of America, Plaintiff-Appellee,
v.
Kevin A. DORTCH, Defendant-Appellant.
No. 09-3260.
United States Court of Appeals, Seventh Circuit.
Argued October 21, 2010.
Decided December 23, 2010.
*924 Thomas S. Ratcliffe (argued), Attorney, Office of the United States Attorney, Hammond, IN, for Plaintiff-Appellee.
J. Curtis Greene (argued), Attorney, Barnes & Thornburg LLP, Indianapolis, IN, for Defendant-Appellant.
Before EASTERBROOK, Chief Judge, and MANION and HAMILTON, Circuit Judges.
MANION, Circuit Judge.
Kevin Dortch robbed a bank in Munster, Indiana, and then led police on two highspeed chases through the nearby residential neighborhoods. He eventually pleaded guilty and was sentenced to the statutory maximum of 240 months' imprisonment. He appeals only his sentence. We affirm.

I.
On September 14, 2007, Kevin Dortch walked through a Jewel-Osco drugstore in Munster into the TCF Bank located inside. He was not wearing a disguise. He handed one of the tellers a note that stated:
I have a gun. Please give me the money out of your drawers. No dye pack. I will be forced to do anything that gets in my way. Nod your head if you understand. You have 30 seconds. Don't push alarm or else.
The teller stacked about $9,000 on the counter and Dortch quickly left the store. Following him outside on the pretext of a smoke break, the bank manager called the Munster Police Department and gave a description of Dortch's blue Ford Escape SUV. Three or four police vehicles responded, and Dortch led them on a high-speed chase through the neighborhoods. The entire chase was caught on video.
The police chased Dortch for several blocks, across the state line into Lansing, Illinois, before Dortch crashed into an attached garage at the end of a cul-de-sac. The SUV engine began to smoke, and Dortch used that as cover to escape over the roof of the single-story garage, leaving the money and the note behind. Dortch fled on foot through back yards until he climbed into an SUV stopped at a traffic light. He ordered the woman inside to drive. Fortunately for everyone involved, the driver was alone in the car and immediately leaped from the vehicle.
Dortch slid into the driver's seat as police officers approached the SUV with guns drawn, and another police SUV pulled in front to prevent Dortch from escaping. Rather than surrender, Dortch rammed the carjacked SUV into the passenger side of the police SUV, knocking it out of the way, and continued his flight for several more blocks. Finally, he lost control of the vehicle turning into an alley and crashed into a detached garage.
Police converged on the garage with weapons drawn. When Dortch refused to come out, the police fired tasers into the garage in an attempt to subdue him. The garage caught fire, but Dortch continued to refuse to leave the garage. Because the police were not yet sure whether anyone else was in the carjacked vehicle, they entered the garage. While attempting to apprehend Dortch and search the garage, one officer, Sergeant Kovacik, cut his hand on glass, necessitating 16 stitches; two others, Officers Huckaby and Cooley, were exposed to potentially dangerous smoke in the burning garage. Officer Huckaby spent two days in the hospital as a result. *925 Dortch eventually emerged from the garage and was arrested.
Dortch was indicted on a single count of bank robbery in violation of 18 U.S.C. § 2113(a) and pleaded not guilty. Eight months later, on the morning his trial was scheduled to begin, Dortch changed his mind and pleaded guilty, without a plea agreement. The district court accepted the plea. The court held three sentencing hearings over a period of four months. The government presented evidence in the form of photographs, video footage, and testimony from one of the police officers involved in the case and an FBI agent. Munster police officer Trisia Fichter testified that Officer Huckaby told her that he had spent two days in the hospital for observation, but Officer Fichter could not identify any tests or treatment performed. The government did not provide any medical records or affidavits from any of the medical staff who encountered Officer Huckaby during his stay.
Based on Dortch's criminal history and numerous sentencing enhancements for his conduct during the robbery and the chase that followed, the district court calculated a guidelines range of 292 to 365 months of imprisonment. The court sentenced Dortch to the statutory maximum of 240 months, noting that he also believed that the maximum sentence was a fair sentence. Dortch appeals his sentence.

II.
Dortch argues that the district court erred procedurally in calculating his guidelines range, but does not challenge the substantive reasonableness of the sentence. Dortch raises three purported errors. First, he argues that the court erred in applying a two-level enhancement for a death threat based on the note he gave the bank teller. Second, he argues that the court erred by enhancing his sentence twice for the same underlying conduct: first, by two levels for his reckless flight and second, by an additional six levels for assaulting a police officer by ramming a police vehicle during the same flight. Third, he argues that the hearsay testimony from Officer Fichter regarding Officer Huckaby's stay in the hospital was insufficient to support a four-level increase for "serious bodily injury" as defined in the guidelines.
We review the district court's application of the guidelines de novo, and its underlying factual findings for clear error. United States v. Munoz, 610 F.3d 989, 992 (7th Cir.2010). Dortch's first two arguments merit little discussion. We have held that the mere statement that a bank robber has a gun can, in context, constitute a threat of death, and here there was considerably more. United States v. Carbaugh, 141 F.3d 791, 794 (7th Cir.1998). And applying multiple enhancements based on the same flight from police is perfectly permissible so long as the enhancements relate to different aspects of the defendant's conduct. United States v. White, 222 F.3d 363, 376 (7th Cir.2000). Here, Dortch received one enhancement for his initial high-speed chase, another for carjacking the second car, and a third for assaulting the police officer blocking his second escape attempt.
The final issue does raise some questions. Application note 1 in the commentary to Sentencing Guidelines § 1B1.1 defines "serious bodily injury" in part as "injury ... requiring medical intervention such as surgery, hospitalization, or physical rehabilitation." The government argues that under this text it only needs to show that someone was "hospitalized" in order to establish a "serious bodily injury," and that Officer Fichter's testimony establishes this. Dortch contends that Officer Fichter's testimony is "unreliable hearsay" *926 and that even if believed it is insufficient to establish serious bodily injury.
We are not concerned with reliability: there is nothing inherently unreliable about Officer Fichter's testimony that Officer Huckaby told her that he spent two days in the hospital because of smoke inhalation. Officer Fichter's testimony established that Officer Huckaby was at the hospital for precautionary observation. But did it establish that there was an actual injury, rather than merely the risk of an injury, from Officer Huckaby's smoke inhalation? And even if there was an injury, did Officer Fichter's testimony establish that a physician's evaluation of the injury, and not some city policy or collective bargaining agreement, was the reason for the overnight stay in the hospital?
This court has not addressed whether hospitalization for precautionary observation because of the risk of serious injury, rather than because of an actual injury, is sufficient for serious bodily injury. Fortunately, we do not need to resolve the issue here on this rather sparse record because any error would ultimately be harmless in that it "did not affect the district court's selection of the sentence imposed." United States v. Anderson, 517 F.3d 953, 965 (7th Cir.2008) (quoting Williams v. United States, 503 U.S. 193, 203, 112 S.Ct. 1112, 117 L.Ed.2d 341 (1992)). No one seriously disputes that there was at least one "bodily injury" in this casethe uncontested evidence that Sergeant Kovacik had lacerations on his hand requiring 16 stitches establishes as much. Even if the government could not, in the end, establish Officer Huckaby's serious bodily injury, Dortch's offense level under the guidelines would only drop by two (the difference between the bodily injury and serious bodily injury enhancements). This would result in a guidelines range of 235 to 293 months. As even Dortch's counsel acknowledged at oral argument, this reduction would not have affected the district court's sentence in any way: the success of the appeal depended on the cumulative effect of his other, unsuccessful, arguments. The district court noted that it felt that the 240-month statutory maximum sentence was a fair sentence in this case, and there is no indication that the fact that this may have been toward the low end of, rather than below, the proper guidelines range would have made any difference. We are confident that, were we to remand this case for resentencing, the district court would impose the same sentence regardless of whether it found that Officer Huckaby suffered serious bodily injury.
Even though any error is harmless, it is a situation that could have easily been avoided. Nothing in the record discloses the reason for Officer Huckaby's stay in the hospital and the treatment he received while there. The government could have easily answered this question with copies of relevant medical records or even a simple affidavit from the treating physician indicating that he ordered Officer Huckaby's hospitalization on account of injuries sustained due to smoke inhalation. Even if Officer Fichter's testimony were enough to support the district court's finding on clear error review, the government should not expect this court to rely on such weak evidence when conclusive proof is so readily available.

III.
Because any error below was harmless, the sentence is AFFIRMED.
