First are related definitions from my lecture note:
A $\textbf{filtration}$ on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is a sequence $(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ of sub-sigma fields of $\mathcal{F}$ such that $\mathcal{F}_{n} \subseteq \mathcal{F}_{n+1}$ for all $n$.
A $\textbf{stochastic process}$ is a collection of random variables defined on the same probability space.
A stochastic process $(X_{n})_{n \in \mathbb{Z}_{+}}$ is $\textbf{adapted}$ to the filtration $(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ if $X_{n}$ is $\mathcal{F}_{n}$-measurable for all $n$.
A process $(X_{n}, \mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ is a $\textbf{martingale}$ if
$(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$ is a filtration and $(X_{n})_{n \in \mathbb{Z}_{+}}$ is adapted to $(\mathcal{F}_{n})_{n \in \mathbb{Z}_{+}}$.
$X_{n}$ is integrable for all $n$.
$\mathbb{E} [X_{n+1} | \mathcal{F}_{n}]=X_{n}$ for all $n$.
Then I have the following exercise:
It's implicitly understood that all random variables are real-valued.
I try to prove that $V_n$ is integrable by looking at the summand $H_{k} (X_{k}-X_{k-1})$. Because $H_{k} (X_{k}-X_{k-1})$ is measurable, it suffices to show that $H_{k} (X_{k}-X_{k-1})$ is bounded. Unfortunately, I can not see how to prove this claim from the facts that $(X_{n})_{n}$ is a $\mathcal{F}_n$-martingale and $(H_{n})_{n}$ is $\mathcal{F}_n$-predictable.
Could you please leave me some hints to finish it? Many thanks!