[Math] Why is this sequence of equicontinuous functions uniformly bounded

Question

Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$. If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded?

Answer 1

Since $[0,1]$ is compact, $f_n$ is uniform continuous. So given $\epsilon>0$, there is a $\delta'>0$ such that for any $x,y\in [0,1], \:|x-y|<\delta'$ $$ |f_n(x)-f_n(y)|<\epsilon\tag1 $$ Since $f_n$ is equicontinuous, $(1)$ holds for all $n$.

Take $\delta=\delta'/2$. For any open cover on $\bigcup_{x\in [0,1]}(x-\delta, x+\delta)$ on $[0,1]$, there is a finite cover $\bigcup_{i\in \{1, \cdots, l\}}(x_i-\delta, x_i+\delta)$ covers $[0,1]$ because it is compact. Since $|f_n(0)|<M'$ and assume $0\in (x_1-\delta, x_1+\delta)$, by $(1)$ for any $x\in (x_1-\delta, x_1+\delta)$ and any $n$ $$ |f_n(x)|<|f_n(0)|+\epsilon<M'+\epsilon $$ Similarly for any $x\in (x_2-\delta, x_2+\delta)$ and any $n$ $$ |f_n(x)|<|f_n(y)|+\epsilon<M'+2\epsilon $$ where $y\in (x_1-\delta, x_1+\delta)$. Repeat this process and we have, for any $x\in (x_l-\delta, x_l+\delta)$ and any $n$ $$ |f_n(x)|<|f_n(y)|+\epsilon<M'+l\epsilon $$ where $y\in (x_{l-1}-\delta, x_{l-1}+\delta)$.

Take $M=M'+l\epsilon$. Then $|f_n(x)|<M$ on $[0,1]$ for any $n$. So $f_n$ is uniform bounded.