In probability, a sequence is said to converge almost surely if the set $\{ \omega \in \Omega : X_n(\omega) \to X(\omega)\}$ has measure one.
In analysis, a sequence $(a_n)_{n \ge 0}$ is said to converge to the limit $a$ if $\forall \epsilon > 0$ there exists $N \in \mathbb{N}$ such that $n \ge \mathbb{N} \Rightarrow |a_n - a| < \epsilon$. A corollary to this definition is that all convergent sequences are bounded in $\mathbb{R}$.
I was wondering if this result could be extended. That is, if you have a sequence of random variables which converge almost surely, are they also bounded on a set with measure one?
My intuition says yes because if you fix an $\omega$ - $X_n(\omega)$ becomes a sequence of real numbers. If each $X_n(\omega)$ is bounded by a real number, then I don't see why $\{\omega \in \Omega: X_n(\omega) \le K\}$ for some $k \in \mathbb{R}$ would not have probability one.