Here is the problem as it is written in the text:

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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 .

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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.

Please help!