Let $X_{n}, n \in \mathbb N_{0}$ be a submartingale where $\sup X_{n} < \infty$. Further, let $\xi_{n}=X_{n}-X_{n-1}$ where $E(\sup \xi_{n}^{+})< \infty$.
Show that $X_{n}-$a.s converges.
This is clearly a case for the martingale convergence theorem, i.e. I need to show that $\sup E(X_{n}^{+})< \infty$
My idea: Since $X_{n}$ is a submartingale, we know that $E(X_{n+1})\geq E(X_{n})\Rightarrow E(X_{n+1}-X_{n})\geq 0$ and from $\sup_{n}X_{n}< \infty$ we know that for an $N \in \mathbb N$ large enough: for all $n \geq N$: $\xi_{n}^{+}=0$
Hence $E(\sup \xi_{n}^{+})=E(\sup\limits_{1\leq n < N} \xi_{n}^{+})$ and
$\sup E(X_{n}-X_{n-1})\leq E(\sup \xi_{n}^{+})$
I do not know how to use these facts to show that $\sup E(X_{n}^{+})< \infty$, any ideas?