Let $f_n, f \in W^{1,p}([0,1])$ for some fixed $p \in (1,\infty)$ and all $n \in \mathbb{N}$. By modification on null sets, we may then assume that $f_n, f \in C[0,1]$.
Now, if $f_n(0)=f(0)$ for all $n \in \mathbb{N}$ and $f_n' \to f'$ in $L^p[0,1]$, is it necessarily true that \begin{equation} f_n \to f \text{ in } W^{1,p}[0,1] ? \end{equation}
I strongly suspect so, but cannot find a way to justify this guess rigorously. Could anyone please help me?
Observe that $f' \in L^p([0,1])$ implies that $f' \in L^1([0,1])$, hence $f(0) + \int_0^x f'(t) \ \mathrm{d} t$ is an absolutely continuous representative of $f$. Knowing this, we have by Hölder's inequality
$$ \begin{align*} |f_n(x) - f(x)| &\le \int_0^x |f_n'(t) - f'(t)| \ \mathrm{d}t \\ &\le \int_0^1 |f_n'(t) - f'(t)| \ \mathrm{d}t \\ &\le \Vert f'_n - f' \Vert_{L^p} \end{align*} $$
So, $$ \int_0^1 |f_n(x) - f(x)|^p \le \Vert f'_n - f' \Vert^p_{L^p} \to 0. $$