I encountered a problem when reading A course in probability theory by Kailai Chung:
If the sequence of ch.f.'s $\{f_n\}$ converges uniformly in a neighborhood of the origin, then $\{f_n\}$ is equicontinuous, and there exists a subsequence that converges to a ch.f. [HINT: Use Ascoli-Arzela's theorem.]
I tried to estimate as follows:
$|f_n(t)-f_n(s)| \le |f_n(t)-f(t)| + |f(t)-f(s)| + |f(s)-f_n(s)|$, where $f$ denotes what $\{f_n\}$ converges into. The first and the third items can be controlled when $n$ is large enough by the uniformly convergence. But I have no idea about the properties of $f$.
The approach above also has another problem, as we only have information in a small neighborhood near $0$.
I know the main reason I cannot figure it out is that I haven't use any properties of characteristic functions. But how? Eagerly anticipating any assistance.