Limit of Cumulative Distribution Function at Positive Infinity

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X$ be the cumulative distribution function.

Then:
 * $\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$

where $\ds \lim_{x \mathop \to \infty} \map {F_X} x$ denotes the limit at positive infinity of $F_X$.