Almost sure inequality and the supremum

Question

I've been reading about convergence lately and I've found that most books assume the following without explaining:

Let $X_n$ be a sequence of random variables, if $|X_n|\leq Y$ almost surely for all $n$ then $sup\phantom{.}_n \phantom{a} |X_n|\leq Y$ almost surely too.

Can someone give a quick proof about it?

Answer 1

For each $n$, let $A_n=\{ \omega : \vert X_n(\omega)\vert \leq Y(\omega)\}$. By hypothesis we know that the probability of $A_n$ is $1$ for each $n$. Then consider $$ A= \bigcap_{n\in\mathbb{N}} A_n $$ $A$ is still a measurable set of probability $1$ (by standard properties of the measure, the complementary is a countable union of sets of probability $0$) and for each $\omega\in A$ we have $\vert X_n(\omega)\vert\leq Y$ for all $n$, so also $\sup_n \vert X_n(\omega)\vert \leq Y$.