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Assume that $\{ X_{n}(\omega)\}$ is a sequence of integer valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $X_{n}\overset{a.s.}{\to} c$, where $c$ is some integer constant. From the definition of a.s. convergence it follows: $$ \mathbb{P}[\omega\in\Omega: \exists \, n_{1}(\omega) \geq 1, \text{such that}, X_{n}(\omega) = c\}, \forall n > n_{1}(\omega) ] = 1 $$

Assume $N(\omega) = \min \{n_{2} \geq 1: X_{n}(\omega) = c, \forall n > n_{2}\}$.

It is always true that $\mathbb{P}[N < \infty] = 1$?

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If $N(\omega)=\infty$, then $\vert X_n(\omega)-c\vert\geq 1$ infinitely often because $X_n$ takes integer values. Thus

$$ \{ \omega: N(\omega)=\infty \}\subseteq \{ \omega: X_n(\omega)\overset{n\rightarrow\infty}{\not\rightarrow} c \} $$

And the RHS has probability $0$.

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