Let $H$ be the vector space of all absolutely continuous and $L^2$-integrable functions on $\Bbb R$ whose derivatives are also $L^2$-integrable. Equip $H$ with the norm $$\|f\|:=\sqrt{\|f\|_2^2+\|f'\|_2^2},$$ where $\|\cdot\|_2$ denotes the $L^2$-norm. Prove that $H$ is a Banach space.
My attempt: pick arbitrarily a Cauchy sequence $\{f_n\}\subset H$, it is then obvious that both $\{f_n\}$ and $\{f'_n\}$ are Cauchy sequences in $L^2(\Bbb R)$, thus there exist $f,g\in L^2(\Bbb R)$ such that $f_n\to f$ and $f'_n\to g$ in $L^2(\Bbb R)$. I think then I will have to show that $f'=g$ a.e. but I'm completely at a loss how to proceed.
Thanks advance for any help.
Comments under this previous post suggest that $H$ is actually a Soblev space $W^{1,2}$, and hence may be handled specifically with Soblev theory. But I'm afraid that would be too advanced for me.
If there really doesn't exist any short and elementary proof, then any reference to an elementary proof is also welcome. Thanks!