I am trying to understand the following explanation: prove/disprove: the functions $f_n(x)=\cos(x+n)+\ln\left(1+\frac{\sin^2(n^nx)}{\sqrt{n+2}}\right)$ are uniformly equicontinuous
Particularly the step in which the accepted answer states that uniform convergence to zero on a compact set implies, by Arzela-Ascoli, the equicontinuity of the family.
I am working from the Pugh textbook in which the follow are stated about Arzela-Ascoli:
Arzela-Ascoli Theorem Every bounded equicontinuous sequence of functions in $C^0([a, b], \Bbb R)$ has a uniformly convergent subsequence.
Arzela-Ascoli Propagation Theorem Pointwise convergence of an equicontinuous sequence of functions on a dense subset of the domain propagates to uniform
convergence on the whole domain.
Corollary Assume that $f_n : [a, b] → \mathbb R$ is a sequence of differentiable functions
whose derivatives are uniformly bounded. If for one point $x_0$, the sequence $(f_n(x_0))$ is
bounded as $n → ∞$ then the sequence $(f_n)$ has a subsequence that converges uniformly
on the whole interval $[a, b]$.
As far as I can tell the Arzela-Ascoli theorem does not imply that step used in that solution. Now on Wikipedia I see that Arzela-Ascoli has a converse, that if every subsequence converges uniformly then the family is equicontinuous. Is this what that answer refers to?
If so, is there some other way to solve that problem which uses only the theorems provided in Pugh's text? The problem is an exercise at the end of this chapter in the textbook, so it seems like it should have a solution which only exercises those ideas.
Best Answer
Arzela _Ascoli Theorem is not required. $f_n \to 0$ uniformly and each $f_n$ is continuous and periodic. This is enough to prove uniform equi-continuity:
There exist $N$ such that $|f_n(x)-f_n(y)|<\epsilon$ for all $x, y$ for all $n >N$. For $1 \leq n \leq N$ the function $f_n$ is continuous and periodic. This implies that it is uniformly continuous. Hence there exists $\delta >0$ such that $|f_n(x)-f_n(y)|<\epsilon$ for all $x, y$ with $|x-y| <\delta$ for all $n \in \{1,2...,N\}$. Can you finish?