Let $h:\mathbb{R} \to\mathbb{R}$ be a continuously differentiable function.
Show that $I_t:=\int_0^t h'(u)B_u du$ has finite variation, where $B_u$ is Brownian motion.
To prove:
$\sup_{N\in \mathbb N} \left\{\sum_{k=1}^N|I_{t_k}-I_{t_{k-1}}|,a\le t_1<\dots<t_N\le b \right\}<\infty$.
Every estimation i take leads to the variation of Brownian motion which is infinite. Can anyone help me?