Prove Arzelà–Ascoli theorem step by step

Prove Arzelà–Ascoli theorem step by step

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0:00 start
5:00 uniformly bounded and continuous need not have converge subsequence
8:00 lemma1, there exist converged subsequence on countable set E
15:00 lemma3, compact metric space has countable dense subset E
19:00 prove interval version of Arzelà–Ascoli theorem
29:00 prove general Arzelà–Ascoli theorem

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We proved that under uniform convergence, one can prove that the limit and the differential can be exchanged.

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