Let $X_n$ be s submartingale and $\zeta_n = X_n - X_{n-1}$ with $\text{sup}\ X_n < \infty$ and sup $\mathbb{E}[\zeta_n^+] < \infty$. Show $X_n \rightarrow X$ a.s.
For some $\epsilon > 0$, let $A_n$ be the event that $\zeta_n > \epsilon$. Since $\text{sup}\ X_n = M < \infty$, and $\mathbb{E}[X_{n+1}] \geq X_n$, then $\mathbb{P}(A_{n+1}) \leq \mathbb{P}(A_{n})$, and $\mathbb{P}(A_{n}) \rightarrow 0$.
Is there any way I can show the sum of probabilities is $0$ and apply Borel-Cantelli? Am I going in the right direction?