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.
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.
This shows that the sequence is uniformly bounded.
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.