Let $f_n\colon \Omega \to \mathbb{R}$ be a sequence of non-negative functions defined on a bounded domain that are increasing a.e. $$f_{n+1} \geq f_n$$ and $$\lVert f_n \rVert_{H^1(\Omega)} \leq C$$ for all $n$.
Hence $f_{n_k} \rightharpoonup f$ weakly in $H^1$ and strongly in $L^2$.
Is there any chance that the increasing property of the functions will imply that the weak and strong limits above apply for the full sequence $f_n$? Unfortunately, I Don't have an upper bound for the $f_n$.
Yes, the full sequence converges (weakly in $H^1$ and strongly in $L^2$).
By the monotonicity of the sequence, it is pointwise convergent (possibly the limit function attains the value $+\infty$ at some points, but that doesn't matter), call the pointwise limit $f$. Then $f$ is the only candidate for a limit of a subsequence, and since each subsequence of $(f_n)$ has a further subsequence that converges, it follows that all subsequences converge to $f$.