Suppose $\{f_n\}_{n\geq1}$ is a series of bounded variation functions on $[a,b]$ satisfying: $$ T_a^b(f_n)\leq M_1, | f_n(a) |\leq M_2 $$ for all $n$, where $M_1, M_2$ are real numbers.
I'm wondering: if $f_n$ converges to a function $f$ everywhere, does it imply
$$ T_a^b(f_n) \rightarrow T_a^b(f)$$
If it doesn't hold, can we make a few justifications?
It just occurred to me when I'm studying real analysis. Although it seems intuitive, I can't directly think of a proof or give a counterexample.
Thanks!