Is the $L^1$ limit of bounded functions again bounded?

Question

For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$. Assume $f_n\to f$ in $L^1(\mathbb{R})$. Is it true that $|f|\leq 1$ almost everywhere ?

I would do it like this: If $f_n\to u$ in $L^1(\mathbb{R})$ we can pass to a subsequence so that $f_{n_k}\to f$ a.e. and hence $|f|\leq 1$ a.e..