Limit of the cumulative distribution function at infinity

Statement

Let $X$ be a real random variable with cumulative distribution function $F_X$. Then $$\lim_{x \to +\infty} F_X(x) = 1.$$

Remarks

The proof: the events $\{X \le n\}$ increase with $n$ and their union is $\Omega$. So continuity from below gives $F_X(n) \to 1$, and $F_X$ is nondecreasing. In the same way, $\lim_{x \to -\infty} F_X(x) = 0$, because the events $\{X \le -n\}$ decrease to $\varnothing$. The proof uses that $X$ takes only real values. For a random variable with values in $[-\infty, +\infty]$, the limit is $1 - P(X = +\infty)$.

Explanations

L1 human

For a very large $x$, it is almost sure that $X \le x$, because $X$ is some real number. So the probability $F_X(x)$ gets as close to $1$ as we like.