An Application using Arzela-Ascoli Theorem I am looking for a good application for undergraduates (Analysis I) that uses the AA-theorem. Perhaps, a proof of some surprising result that is not so easy to prove otherwise? I do realize that AA-theorem is usually used as a theoretical tool in other proofs. But I was wondering if anyone has any insightful examples that may make the students appreciate it? 
 A: I suppose it's probably too difficult to introduce compact operators and the Schauder fixed point theorem at this stage. But if you are willing to use them as black boxes, then you can do a trivial application of Schauder FPT: define the operator $T:C[0,1]\to C[0,1]$ by
$$
Tf(x) = \int_0^x f(t)~dt + 1.
$$
You can prove that the image of the unit ball of $C[0,1]$ under $T$ is pointwise bounded and equicontinuous, so $T$ is a compact operator. Therefore $T$ has a fixed point, which is the solution to the integral equation
$$
f(x) = \int_0^x f(t)~dt + 1,
$$
which is a way to recast the first-order ODE/IVP
$$
f'(x) = f(x), ~f(0) = 1.
$$
This is, of course, the exponential function. While it's a trivial ODE, the idea generalizes easily and it should give some insight into how Ascoli-Arzela is used in practice, as well as the historical development of the subject.
If you don't want to rely on compact operators, you should still be able to solve the ODE using the approach of the Peano existence theorem, which concerns the existence of short-time solutions to the ODE
$$
y'(t) = f(t,y(t))
$$
where $f$ satisfies a Lipshitz condition in $y$. While the full proof of Peano is a bit laborious, adapting the proof to our special case should be enough to illustrate the main ideas: (1) convert to the integral equation, (2) form piecewise linear approximations to the solution, (3) apply Ascoli-Arzela on the set of approximations to extract a convergent subsequence which should be the solution.
If you intend to or have already discussed the contraction mapping principle, this might also be a nice way to tie that discussion together, because the proof of the Picard existence theorem also converts the ODE to an integral equation, this time to obtain a contraction mapping. Again, this should serve to provide historical insights and motivations for these tools, which are used in much the same way even now.
