Let $\{ X_n \}_{n \in \mathbb{N}}$ be a positive valued stochastic process that converges almost surely to an integrable random variable $X_{\infty}$ then is it true that
$$ -\infty < \sum_{n= 1}^{\infty} E[ X_{n+1} - X_n | X^n ] < \infty $$
where $X^n= \{ X_n, X_{n-1}, \dots, X_1 \}$. I know that almost sure convergence does not guarantee convergence in mean but I believe this statement is being used in a proof I am trying to understand, is it true?
EDIT: as pointed out by Marcus M, there are trivial counterexamples if some bounds on the means is not assumed. So I would add the assumption that $E[ X_{n+1} | X^n ] < \infty \ \forall{n \in \mathbb{N}}$ and $E[X_1] < \infty$ to keep the question relevant.
I think you can come up with a super basic that breaks this: let $X_1$ be any random variable with $E[X_1] = +\infty$, and let $X_j = 1$ for all $j \geq 2$.