real analysis - If $f_nto f$ in the $L^1$ norm, show that there is a subsequence $f_{n_k}$ which converges a.e. to $f$.

This question is from a problem set on $L^p$ spaces in my undergraduate-level real analysis course. I said that $f_n$ converges if and only if it is Cauchy. Therefore, $\exists \, N\in\mathbb{N} \; \forall \, m,n>N \; ||f_m - f_n||_1 < \epsilon$. I then stated that we can choose a subsequence that will also be Cauchy; for example, let $f_{n_k}=f_{N+k}$. Since the $f_n$'s are Cauchy, we have that $\forall \, \ell,k \; ||f_{n_k}-f_{n_\ell}|| < \epsilon$. Therefore, $f_{n_k} \to f$ almost everywhere.

This proof is evidently erroneous, as I apparently cannot use the Cauchy condition in this way (I can't even read my grader's handwriting on this problem so I can't be sure of where exactly the inference fails, but I know it has to do with the Cauchy condition). I probably underestimated the difficulty of this problem. How can I fix this proof so it works?