Cool. I knew that Stone's theorem is what you use in the unbounded case, but I have never studied the details. I wasn't aware of the stuff mentioned in the attachment.
To a typical physics student, a rigorous treatment of the bounded case is not trivial. They aren't expected to understand (or even care about) how to prove that the series that defines the exponential is convergent given that H is bounded, or what sort of things they can do to series and other expressions that involve limits. A physics student who's given this problem is expected to answer it with [U(t),H]=\left[e^{-iHt},H\right]=\left[\sum_{k=0}^\infty\frac{(-iHt)^k}{k!},H\right]=\sum_{k=0}^\infty \frac{(-it)^k}{k!}[H^k,H]=0 without even thinking about whether these steps actually make sense.