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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?

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    You can't use the Arzelà–Ascoli theorem; rather, this exercise shows that your hypotheses imply that the hypotheses of the Arzelà–Ascoli theorem would apply. – 2017-02-09
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    We hae that $|f_n(p)|$n\in \mathbb N$ and $x,y\in [0,1],\ $ there is a $\delta >0$ such that $|x-y|<\delta \Rightarrow |f_n(x)-f_n(y)|<1 .$ Cover $[0,1]$ with a finite number, say $N$ of balls of diameter $<\delta$, centered at the points $x_1<\cdots $x\in [0,1].$ Wlog assume $x$x\in B(x_1).$ Then, $|f_n(x)|\le |f_n(x)-f_n(x_1)|+\dots +|f_n(x_m)-f_n(p)|+|f_n(p)| – 2017-02-09

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