Is this characterization of weak differentiablity exact?

Question

I think that $f$ is weakly differentiable if and only if there exists a sequence of smooth functions $f_n$, which converges locally to $f$ in $L^1$ and such that $f'_n$ is locally Cauchy in $L^1$. Then the weak differential is the (local) limit of $f'_n$ in $L^1$.

Is that true?

EDIT: I have deleted my wrong proof.

Answer 1

It appears the theorem is true (if my understanding is exact). Here is a pic of Gilbarg and Trudinger, Elliptic partial differential equations of second order, thm. 7.4