Random variable inequality: $(X \le x)$

Question

$(X \le x) = (X \le x, |Y - X| \le \epsilon) \cup (X \le x, |Y - X| > \epsilon) \subset (Y \le x + \epsilon) \cup (|Y - X| > \epsilon)$

I don't understand this at all. Where does the last $\subset$ come from?

Answer 1

In this case, we can split the inclusion: both

$$(X \leq x, |Y - X| \leq \epsilon) \subset (Y \leq x + \epsilon)$$

and

$$(X \leq x, |Y - X| > \epsilon) \subset (|Y - X| > \epsilon)$$

hold. From this, we achieve the inclusion you're having trouble with. Try to prove both these inclusions separately.

Hint: what can we say about $Y$ is we know that $X$ is at most $x$, and $Y$ cannot differ more than $\epsilon$ from $X$?