There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic Programming" the theorem is:

"Let $Y_t, X_t, Z_t, t = 1,2,3,....$ be three sequences of random variables and let $\mathcal{F_t}$ be sets of random variables such that $\mathcal{F_t} \subset \mathcal{F_{t+1}}$ for all t, suppose that:

(a) The random variables $Y_t, X_t, Z_t$ are nonnegative and are functions of the random variables in $\mathcal{F}_t$

(b) For each $t$ we have $E[Y_{t+1}|\mathcal{F_t}] \leq Y_t - X_t +Z_t$

(c) $\sum_{t=0}^\infty Z_t \lt \infty$

Then:

$\sum_{t=0}^{\infty}X_t \lt \infty $ and there exists a nonnegative random variable $Y$ such that $Y_t \rightarrow Y$ with probability 1."

The problem is, I can't find any proofs of this anywhere. Most texts on probability theory prove the standard Martingale Convergence Theorem but I feel this is a big step from that.

Can anyone direct me to a source which proves the above or write out a proof from the standard convergence theorem?