If $f_n \rightharpoonup f$ in $L^2[0,1]$, then $F_n\to F$ uniformly where $F_n=\int_{0}^{x}f_n$

Question

This is the full problem. a) Is easy. However, I feel I solved b) not in an intended way so I would want to get a check on my work and perhaps some different solutions.

b) This is simply weak convergence and so the sequence $f_n$ is bounded in $L^2$. We also have pointwise convergence by weak convergence. Now I claim $F_n$ are equicontinuous. $|\int_{y}^{x}f_n|\leq\|f_n\|_2|x-y|\leq M|x-y|$ Where $M$ is the bound on $f_n$. $[0,1]$ is compact so equicontinuity + pointwise implies uniform.