0

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?

eepperly16
  • 7,232
score324
  • 177
  • 7
  • 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. – Jonas Meyer Feb 09 '17 at 05:27
  • 1
    We hae that $|f_n(p)|<M<\infty,\ $ and we also know that for all $n\in \mathbb N$ 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 $x_1<\cdots <x_N.$

    Let $x\in [0,1].$ Wlog assume $x<p$ and $x\in B(x_1).$ Then,

    $|f_n(x)|\le |f_n(x)-f_n(x_1)|+\dots +|f_n(x_m)-f_n(p)|+|f_n(p)|<N+M.$

     – Matematleta Feb 09 '17 at 06:03

0 Answers0