Monotonicity of the cumulative distribution function

Statement

Let $X$ be a real random variable with cumulative distribution function $F_X$. Then $F_X$ is nondecreasing: if $x \le y$, then $F_X(x) \le F_X(y)$.

Remarks

The proof: $\{X \le x\} \subseteq \{X \le y\}$, so the claim follows from the monotonicity of probability. More precisely, $F_X(y) - F_X(x) = P(x < X \le y) \ge 0$. A nondecreasing function has at most countably many points of discontinuity. So $P(X = x) > 0$ for at most countably many $x$.

Explanations

L1 human

As $x$ moves to the right, the event "$X \le x$" contains more outcomes, so its probability cannot go down.