Convergence a.e. but not in $L_{p}$ norm Corollary

Question

Let $p>1$. I have considered the following example of a function in $L_{p}(\mathbb{R})$ that converges to $0$ almost everywhere but doesn't converge to $0$ in the $L_{p}$ norm: for each $n$, let $f_{n}=n\chi_{(0,\frac{1}{n}]},$ where $\chi_{(0,\frac{1}{n}]}$ is the characteristic function of ${(0,\frac{1}{n}]}$.

Now, for any $\epsilon >0$ I need to find a measurable subset $E_{\epsilon}$ of $[0,1]$ such that $m(E_{\epsilon})<\epsilon$ and $f_{n}(x) \to 0$ uniformly on $[0,1] \setminus E_{\epsilon}$. I am stuck finding this subset, even though it should probably be straightforward, given that the function is quite simple.

Answer 1

Take $E_{\epsilon}=(0,\epsilon /2)$. Note that $x \notin E_{\epsilon}$ imlies $f_n(x)=0$ for all $n>\frac 2 {\epsilon}$.