Weak derivative and approximation

Question

Let $f \in L^2(\mathbb R)$ be absolutely continuous. Is it true that if $f_n$ is a sequence of Lipschitz functions s.t. $f_n \rightarrow f$ in $L^2$ and $ f_n' \rightarrow g $ in $L^2$ then $ \|g\|_{L^2} \geq \|f'\|_{L^2}$?

Answer 1

This is true in even weaker hypothesis, i.e. only weak convergence. In fact, taking $\phi \in C_c^{\infty}(\mathbb R)$, we have $$ \begin{matrix} \int f'_n \phi & = & - \int f_n \phi'\\ \downarrow & & \downarrow\\ \int g \phi & & -\int f \phi' &= \int f' \phi \end{matrix} $$ therefore $f'=g$ a.e..