Convergence almost uniformly implies convergence in $L_\infty$

Question

Show that if $f_n \rightarrow f$ almost uniformly then $f_n \rightarrow f $ in $L_\infty$.

I know that convergence in $L_\infty$ implies converge uniformly and hence almost surely but I have problems to show the converse, because I not see how construct sets of 0 measure such that in our complements the $|f_n(x)-f(x)|< \varepsilon$.

Any hint or help I will be very grateful.

Answer 1

This is false. $\chi_{(0,\frac 1 n)}$ is a counter-example.