Is the proof even valid? The claim is that the integral (1) is bounded between 0 and a positive number decaying to 0 as n tends to infinity, which sounds like a contradiction.

But the integral (1) is itself a function of n, since both f and F are. Not a fixed number. So the inequality is eminently satisfiable.

In other words, if we call (1) I, they show

0 < I < π^n a^n / n!

which is bad. But I depends on n so this really should read

0 < I_n < π^n a^n / n!

which is fine... as far as the proof shows.

But perhaps I'm missing something...