Here is the problem as it is written in the text:
Let be defined and continuous for and . The purpose of this exercise is to show that the problem , , has a solution on the interval for some . Perform the operations as follows: Divide into equal parts: , and define a continuous function inductively by
Put , so that
Use the Arzela-Ascoli Theorem to find a convergent subsequence of the . Show that the limit satisfies and .
The version of the A-A Thm that seems the most applicable here is the following:
Let be a compact set. If is (uniformly) equicontinuous and pointwise bounded, then every sequence in has a uniformly convergent subsequence.
Letting , I don't know how to show equicontinuity or pointwise boundedness. I know that each individual is bounded and uniformly continuous, but that doesn't mean that they're pointwise bounded or equicontinuous (because there are infinitely many ).
EDIT: I've figured out what to do after applying the A-A Theorem. However, I still don't know how to prove the criteria for using the A-A Theorem, so I still need help with that.
Also, I'm not sure where that constant came from. It pops up and then disappears almost immediately, never to be mentioned again. Maybe it's a typo.