# a.s. convergence to constant

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 $$\mathbb{P}[N < \infty] = 1$$?

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 $$0$$.