The question I have in mind is the following: how can we prove that for any $v\in H^1(\Omega)$ and for any $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ the Gauss-Green identity takes place

$$\int_{\Omega} v(x)\Delta u(x)\ dx=\int_{\partial\Omega}\frac{\partial u}{\partial\nu}(x) v(x)\ d\sigma-\int_{\Omega}\nabla u(x)\cdot\nabla v(x)\ dx \;\;? $$

Here $\Omega\subset\mathbb{R}^N$ is a bounded Lipschitz domain: in some books I found this identity assuming that $u\in H^2(\Omega)$ (like Atkinson - Theoretical Numerical Analysis, page 325).

I tried myself to prove this, using the weak form of Gauss-Green formula, and I came across the following simple-looking fact that I do not know how to prove.
Let $h\in H^1(\Omega)$ be such that there exists $\Delta h\in L^2(\Omega)$ ($h$ may not be in $H^2(\Omega)$) and consider a sequence $(f_n)_n\subset C^{\infty}(\Omega)$ with $f_n\to h$ in $H^1(\Omega)$.
My question is: is it true that $$\lim\limits_{n\to\infty} \int_{\Omega} g(x)\Delta f_n(x)\ dx=\int_{\Omega} g(x)\Delta h(x)\ dx$$

for all $g\in H^1(\Omega)$? Knowing this is true will suffice for our aim.