Doob's decomposition and submartingale with bounded increments Let $(X_n)_{n \geq 0}$ be a submartingale defined on some filtered probability space $(\Omega, \mathcal{F}, ({\mathcal{F}}_n)_{n \geq 0}, \mathbb{P})$. It is a standard fact that $X_n = X_0 + M_n + A_n$, where $(M_n)_{n \geq 0}$ is a martingale null at $0$ and $(A_n)_{n \geq 0}$ is an increasing previsible process, also null at $0$.
Suppose that there exists a positive constant c such that $|X_{n+1} - X_n| \leq c$ for all $n \geq 0$. Then we need to prove the following:
\begin{equation}
\bigg\{ \sup_{n \geq 0 } X_n < + \infty \bigg\} \subseteq \bigg( \big\{ (X_n) \text{ converges in } \mathbb{R} \big\} \cap \big\{A_\infty < +\infty \big\} \bigg).
\end{equation}

Remember that when you prove the Doob's decomposition theorem you define:
$$A_n = \sum_{k=1}^n \mathbb{E}[X_k | \mathcal{F}_{k-1}]-X_{k-1}$$
and
$$M_n =X_0+ \sum_{k=1}^n X_{k}-\mathbb{E}[X_k | \mathcal{F}_{k-1}]$$
Being $X_n$ a submartingale, you have $A_n \ge 0$ this implies $M_n \le X_n$ which implies $\sup_nM_n < + \infty$ if we are in the event $\{\sup_nX_n < \infty \}$.
Now $M_n$ is a bounded martingale then it converges. And we have $A_n \le 2\sup X_n -\inf M_n < \infty$ being $A_n$ increasing you have the result.
