What does it mean to say that $\left(Y_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0}$ is a supermartingale?

State and prove Doob's Upcrossing Inequality for a supermartingale.

Let $\left(M_{n}, \mathcal{F}_{n}\right)_{n \leqslant 0}$ be a martingale indexed by negative time, that is, for each $n \leqslant 0$, $\mathcal{F}_{n-1} \subseteq \mathcal{F}_{n}, M_{n} \in L^{1}\left(\mathcal{F}_{n}\right)$ and $E\left[M_{n} \mid \mathcal{F}_{n-1}\right]=M_{n-1}$. Using Doob's Upcrossing Inequality, prove that the limit $\lim _{n \rightarrow-\infty} M_{n}$ exists almost surely.