Suppose $\{X_n\}_{n = 1}^{\infty}$ is a discrete-time submartingale (a sequence of random variables, such that $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$), such, that $\forall n \in \mathbb{N} EX_n^2 < \infty \text{ and } P(X_n > 0) = 1$. Is it true, that $P(\exists \lim_{n \to \infty} X_n) = 1$?
I know, that as $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$, and, by Jensen inequality for conditional expectations $P(E[X_{n+1}^2|X_1, … X_n] \geq E^2[X_{n+1}|X_1, … X_n]) = 1$, we have that this question can be solved by proving one of the following statements: $$P(\exists \lim_{n \to \infty} E[X_{n+1}|X_1, … X_n]) = 1$$ $$P(\exists \lim_{n \to \infty} E[X_{n+1}^2|X_1, … X_n]) = 1$$ However, they do not seem to be any easier to prove.
A simple counter example is $X_n=n$ for all $n$.