I have a sequence of continuous functions $f_{n}:[0,1]\rightarrow\mathbb{R}$ that converge pointwise to $0$ on $[0,1]$. I have to prove that given $\epsilon>0$ I can find a non-empty interval $(a,b)\subset [0,1]$ and N such that:
$|f_{n}(t)|<\epsilon$ for all $t\in(a,b)$ and $n\geq N$.
My attempt:
I look at $E_{N}=\{x\in[0,1]:|f_{n}(t)|<\epsilon, \forall n\geq N\}$. Then $E_{N} \subset E_{N+1}$. Forthermore, as $f_{n}$ converges pointwise to $0$, we have that for any $x$ in $[0,1]$ there exists some $n$ such that $x \in E_{n}$. So we must have that $\cup_{N=0}^{\infty} E_{N}=[0,1]$.
I don't know how to conclude that there exists such an $(a,b)$.