Submartingale convergence (Durrett 5.3.1)

You've stepped off a cliff in deducing that the expectation $E(X^+_n)$ is bounded above by the random variable $\sup_n X_n$.

Suggestion: Employ the argument used by Durrett in the proof of his Theorem 5.3.1. Fix a positive real $K$, define the stopping time $T=T_K$ to be the first time $n$ that $X_n$ is larger than $K$, and observe that the stopped process satisfies $$ X_{n\wedge T}\le K+\sup_m\xi_m^+, $$ so that $$ E(X_{n\wedge T})\le K+E(\sup_m\xi_m^+)<\infty,\qquad\forall n. $$ Now apply the submartingale convergence theorem to the stopped process. This yields a.s. convergence of $X_n$ on the event $\{T_K=\infty\}=\{\sup_mX_m\le K\}$. Finally, vary $K$.