I want to solve the following exercise from Durrett:
"Let $X_n, n\geq0$ be a submartingale with $\sup X_n<\infty$. Let $\xi_n = X_n - X_{n-1}$ and suppose $\mathbb{E}\left(\sup\xi^+\right)<\infty$. Show that $X_n$ converge a.s. "
I am not sure to understand the hypothesis on the increments because I have the feeling that I can solve the exercise without it : $\sup X_n<\infty$ implies that we can find $K$ such that for all integer n $X_n<K$ thus we can consider the submartingale $Y_n = X_n - K$ and since $\sup\mathbb{E}(Y_n^+)=0 < \infty$ by the Martingale convergence theorem $Y_n$ converge a.s and thus $X_n$ too.
Is this correct ?