Given a sequence $f_n:[0,1]\to[0,1]$ of continuous functions, define $g_n:[0,1]\to{\Bbb R}$ by setting $$ g_n(x)=\int_0^1\frac{f_n(t)}{(t-x)^{1/3}}dt,\quad x\in[0,1]. $$ Show that $(g_n)_{n\in{\Bbb N}}$ has a uniformly convergent subsequence.
Since $K=[0,1]$ is compact, by Arzela-Ascoli, if one can show
- $g_n\in C(K)$;
- $\{g_n\}$ is pointwise bounded;
- $\{g_n\}$ is equicontinuous on $K$;
then we are done.
I have difficulty in 3. I messed up by only using the definition of equicontinuity. Is there any other theorem which might be useful here?