Martingale convergence theorem with alternative assumptions

Question

Has anyone a source (book, paper, ...) containing a proof of the following theorem ?

Let $(X_n)_{n\in\mathbb{N}_0}$ a sequence of nonnegative random variables adapted to a filtration $(\mathcal{F}_n)_{n\in\mathbb{N}_0}$, and satisfying the conditions
(i) $E(X_{n+1}\mid \mathcal{F}_n)\geq X_n$ almost sure for all $n\in\mathbb{N}_0$
(ii) $\sup_{n\in\mathbb{N}_0}E(X_n\mid\mathcal{F}_0)<\infty$ almost sure.
Then $X_n$ conveges almost sure to a finite random variable $X_\infty$