Consider a sequence of equicontinuous functions which are pointwise convergent. Then why they are uniformly convergent on compacts ?
2026-04-03 11:48:17.1775216897
A basic question on uniform convergence on compacts
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Let $f$ be the pointwise limit of the sequance $\{f_n\}$. By the Ascoli-Arzèla theorem, every subsequence of $\{f_n\}$ has a subsequence that converges uniformly. The limit od these unuiformly concverging subsequences must be the same: $f$. This is enough to show that $\{f_n\}$ converges uniformly (argue by contradiction.)