TITLE: Convergence in $L^{\infty}$ norm implies convergence in $L^1$ norm
QUESTION [5 upvotes]: Let $\{f_n\}_{n\in \mathbb{N}}$ be a sequence of measurable functions on a measure space and $f$ measurable. Assume the measure space $X$ has finite measure. If $f_n$ converges to $f$ in $L^{\infty}$-norm , then $f_n$ converges to $f$ in $L^{1}$-norm. 

This is my approach:
We know $||f_n-f||_{\infty} \to 0 $ and by definition $||f_n-f||_{\infty} =\inf\{M\geq 0: |f_n-f|\leq M \}.$ Then
\begin{align}
||f_n-f||_1\
&=\int |f_n-f| dm\
&\leq \int|f_n|dm+\int|f|dm\
\end{align} 
I don't know how to proceed after that, any help would be appreciated.

REPLY [3 votes]: Your mistake is using the triangle inequality when you don't have to. Here,
$$
\|f_n-f\|_1=\int|f_n-f|dm\leq\|f_n-f\|_\infty\,\int1dm=\|f_n-f\|_\infty\,m(X)\to0.
$$