Problem: Let $\{f_k\} \in BV([a,b])$ and $f_k \rightarrow f$ pointwise in $[a,b]$. Suppose $V(f_k;a,b) \le M$ for all $k \in \mathbb{N}$ for some $M \in (0,\infty)$. Is it enough for a pointwise convergence to make $f$ be of bounded variation. Prove or disprove.
Comments: As far as I understood, this problem is the same as the ones in link1 and link2. However, the accepted answers in these links differ, as link1 proves the statement, link2 disproves it. In the last part of the accepted answer in link2, authors states:
Each $f_n$ is of bounded variation because it is decresing and bounded on $(0,1]$.
Which I think is not correct as we are working on closed interval $[a,b]$, not $(a,b]$. So, is it indeed enough for a pointwise convergence to make $f$ be of bounded variation?