Can a Subsequence of Measurable Functions Converge in L1?

[Funky1981]

Homework Statement

Let fn be a sequence of measurable functions converges to f a.e. Is it possible to get a subsequence fnk of fn s.t. fn converges in L1 ?

2. The attempt at a solution
I have proved the converse statement is true and i guess the above statement is impossible but I fail to prove it.

 

[jbunniii]

Hint: Look for a counterexample where $f_n \rightarrow 0$ pointwise but $\|f_n\| = 1$ for all $n$.

 

[Funky1981]

Thanks for your suggestion