For a filtration $\{F_n\}_{n\ge0}$ on the probability space $(\Omega,\mathcal{F},\mathbb{P})$, let $\{M_n\}_{n\ge0}$ be an $\{F_n\}$-submartingale and let $\{H_n\}_{n\ge1}$ be an $\{F_n\}$-predictable process such that each $H_n$ is nonnegative and bounded. Show that the stochastic process $\{Y_n\}$ defined by: \begin{align} Y_0(\omega):=0, \quad Y_n(\omega):=\sum\limits_{k=1}^{n}H_k(\omega)(M_k(\omega)-M_{k-1}(\omega)), \ n\ge1, \ \omega\in\Omega \end{align} is an $\{F_n\}$-submartingale.
Here is what I have so far:
\begin{align} \mathbb{E}(Y_{n+1} \ \big| \ F_n)&=\mathbb{E}\big(\sum\limits_{k=1}^{n+1}H_k(M_k-M_{k-1}) \ \big| \ F_n\big)\\ &=\sum\limits_{k=1}^{n+1}\mathbb{E}\big(H_k(M_k-M_{k-1}) \ \big| \ F_n\big)\\ &=\sum\limits_{k=1}^{n+1}\big[H_k\mathbb{E}(M_k-M_{k-1} \ \big| \ F_n)\big] \quad \text{since for all $k\le n+1$: $H_k \in F_{k-1}\subseteq F_n $}\\ \end{align}
Now I know that $\mathbb{E}(M_{n+1} \ \big| \ F_n)\ge M_n$ but I am not sure how to deal with these quantities $\mathbb{E}(M_k-M_{k-1} \ \big| \ F_n)$ for $k\le n+1$ in the last line of my proof so far. Is there a way to manipulate those quantities and finish off this proof or is there a better approach that one can take? Thanks in advance.
Right, this looks good so far. What you would want to do from here is use that $\mathbb{E}[M_k-M_{k-1}|\mathcal F_n] = M_k-M_{k-1}$ for all $k \le n$, so \begin{align*}\sum_{k=1}^{n+1}(H_k \mathbb{E}[M_k-M_{k-1}|\mathcal F_n]) &= \sum_{k=1}^{n}H_k (M_k-M_{k-1})+ H_{n+1} \mathbb{E}[M_{n+1}-M_{n}|\mathcal F_n] \\ &= Y_n + H_{n+1} \mathbb{E}[M_{n+1}-M_{n}|\mathcal F_n]. \end{align*}
Now you just use the fact $\mathbb{E}[M_{n+1} | \mathcal F_n] \ge M_n$ you mentioned to finish off the proof.