Let $(f_{n})$ be a sequence in $C[0,1]$ that is equicontinuous on $[0,1]$, and let $p\in [0,1]$ be given. Show that if $(f_{n}(p))^{\alpha}_{n=1}$ is bounded, then $(f_{n})$ is uniformly bounded.
Can I use Arzela Ascoli theorem to prove the above problem?