# supermartingale which is closable on the right

Let $\{X_t\}$ be a supermartingale (right continuous) which is bounded from below, i.e. $X_t\ge -a$ for all $t$. Now let $Y_t:=X_t+a$ which is therefore a positive supermartingale. Convergence theorems tells us, that $Y_t\to Y_\infty$ almost surely and $Y_\infty\in L^1$. This is the convergence theorem as I know it. Now why is the following true:

$$E[Y_\infty|\mathcal{F}_t]\le Y_t$$

for all $t\ge 0$, i.e. $\{Y_t\}$ is closable from the right. We would then conclude that the same is true for $\{X_t\}$. It would be appreciated if someone could give me proof why this is true, for simplicity, in the discrete case.

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Fatou's lemma. ${}$ –  Byron Schmuland Sep 29 '12 at 14:08
@ByronSchmuland thx for the answer! –  user20869 Sep 29 '12 at 14:14