Show that if $X$ and $Y$ are random variables, then $\{X \le Y \} $ is an event. I am studying pre-measure probability theory and only the simplest definitions are known.
My approach:
I know that a random variable $X:\Omega \to \mathbb R$ is such that for any $B \in \mathcal{B}(\mathbb R)$, $X^{-1}(B) \in \mathcal{F}$. Where $B$ is a member of the Borel Sets on the real line.
For a single random variable, I could say that for any $t \in \mathbb R$, $\{X^{-1}(B)\} \in \mathcal{A}$, where $\{X^{-1}(B)\} = X^{-1}((-\infty, t]))$, ( so every $X^{-1}$ is an event...). I'm not sure how to extend this to show that $\{X\le Y\}$, is also an event; I feel like I am missing something really simple.
Thanks!