If we have that $X < \infty$, why doesn't it imply $X$ is bounded?

Question

Suppose that $X$ is some sequence or random variable, etc. I am trying to understand why if $X < \infty$, then that doesn't necessarily mean it has a bound, i.e., there doesn't exist some value $M$ whereby $X \leq M$ for all possible realizations of $X$. Does anyone have an intuition why? Thanks

Answer 1

Say $X$ successively takes the values $1,2,3, ...$. Then $X$ is unbounded, even though each value it takes is finite.

Answer 2

Assuming it is not a special ad-hoc notaion, but that it means "$X(\omega)<\infty$ for all $\omega$":

For instance $X:\Bbb R\to\Bbb R$, $X(\omega)=\omega$.