Let $(X,\mathcal{F},\mu)$ be a measurable space, show that $f_n \rightarrow f$ in $L_{\infty}$ if and only if $\{f_n\} \rightarrow f$ uniformly.
I proved $\Rightarrow) $ but in the other direction i have problems i try to use: $$ \mathrm{lim}_{n \rightarrow \infty} || f_n - f||_{\infty} = 0 \Leftrightarrow \forall \varepsilon > 0 \exists N: n > N \Rightarrow |f_n(x) - f(x)| \leq || f_n - f||_{\infty} < \varepsilon$$ $\mu$-almost everywhere on $X$.
But i not see how use aplicated, maybe is a conceptual problem.
Any hint or help i will be very grateful