The Arzela-Ascoli Theorem is a rather formidable-sounding theorem that gives a necessary and sufficient condition for a sequence of real-valued continuous functions on a closed and bounded interval to have a uniformly convergent subsequence.

Statement: Let be a uniformly bounded and equicontinuous sequence of real-valued continuous functions defined on a closed and bounded interval . Then there exists a subsequence that converges uniformly.

The converse of the Arzela-Ascoli Theorem is also true, in the sense that if every subsequence of has a uniformly convergent subsequence, then is uniformly bounded and equicontinuous.

Explanation of terms used: A sequence of functions on is uniformly bounded if there is a number such that for all and all . The sequence is equicontinous if, for all , there exists such that whenever for all functions in the sequence. The key point here is that a single (depending solely on ) works for the entire family of functions.

## Application

Let be a continuous function and let be a sequence of functions such that

Prove that there exists a continuous function such that for all .

The idea is to use Arzela-Ascoli Theorem. Hence, we need to show that is uniformly bounded and equicontinuous.

We have

This shows that the sequence is uniformly bounded.

If ,

Similarly if , .

If and ,

Therefore we may choose , then whenever , . Thus the sequence is indeed equicontinuous.

By Arzela-Ascoli Theorem, there exists a subsequence that is uniformly convergent.

.

By the Uniform Limit Theorem, is continuous since each is continuous.

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