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This seems like a really simple question but I am struggling to prove it; any help would be useful! Perhaps it is not true, in which case, a counterexample would alternatively be very much appreciated!

Let $\{f_n\}_{n=0}^\infty$ be a sequence of real-valued functions on $[0,T]$ such that $\sup_{t \in [0,T]} |f_n(t)| \rightarrow 0$ as $n \rightarrow \infty$. Then is it true that the total variation $\text{TV}(f_n)$ defined as the supremum over all partitions $0 = t_0 < t_1 < \ldots < t_m = T$ of $\sum_{i=1}^m |f_n(t_i) - f_n(t_{i-1})|$ also vanishes as $n \rightarrow \infty$?

Many thanks!

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To build a counter example, divide the interval $[0,T]$ in $2^n$ equal parts. Then define $f_n(x) = 1/n$ on every other interval of size $T/2^n$ and $f_n(x) = 0$ elsewhere (hope this is clear...). You can calculate that TV$(f_n) = 2^n/n$.

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Oh yes sorry this is obvious now I have seen it! Thanks! –  Jon Apr 19 '13 at 14:23

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