In probability books, the definition of independent discrete random variables are often given as
The random variables $X$ and $Y$ are said to be independent if $\mathbb P(X \leq x, Y \leq y) = \mathbb P(X \leq x) \mathbb P(Y \leq y)$ for any two real numbers $x$ and $y$, where $\mathbb P(X \leq x, Y \leq y)$ represents the probability of occurrence of both event $\{X \leq x\}$ and event $\{Y \leq y\}$.
or
$\mathbb P(X \in A, Y \in B) = \mathbb P(X \in A) \mathbb P(Y \in B)$
And the 2 definitions are alleged to be identical. But the proof is often omitted. Although it's intuitively correct, I still want to see a proof. Could anyone show me how to prove this?