Sequence of functions in $L^p$ and derivative

Question

if we have a sequence of functions, $(f_n)_n \subset C^1(\mathbb{R})$ such that $f_n, f'_n \in L^p(\mathbb{R}) $ and $$f_n\rightarrow \phi_0 \, in \,L^p, $$ and $$f'_n\rightarrow \phi_1 \, in \,L^p.$$ Also, we can show that $(f_n)_n $ and $(f'_n)_n $ converge uniformly on $\mathbb{R}$. Can we conclude that $ \phi_0$ and $ \phi_1$ are in fact this uniform limits and in this case, $$ \phi_0 \in C^1(\mathbb{R})$$and $$\phi'_0=\phi_1.$$ Thank you for your help.

Answer 1

Hint:

Use the fact that any sequence that converges in $L^p$ has a subsequence that converges pointwise almost everywhere. Moreover, you know that the sequence $f_n$ in particular converges pointwise by converging uniformly.