Almost sure equality if $P(X_n = x \text{ infinitely often})=1$ presumed

Question

Let's say $(X_n)_{n\in N}$ is a non-negative supermartingale. From the Martingale convergence theorem follows that there exists $lim_nX_n = X_\infty$ a.s.

I presume that $P(X_n = x \text{ infinitely often})=1$ for some $x$.

Is this statement true: $x = X_\infty$ a.s.? If yes, how can I show it?

Answer 1

Just do it for each fixed $\omega$ for which $X_{\infty} (\omega)=\lim X_n(\omega)$. For real sequences $a_n=x$ for infinitely many $n$ and $a_n \to a$ implies $a=x$ right?