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 $F_X$.

Proof
From Sequential Characterisation of Limit at Positive Infinity of Real Function: Corollary, we aim to show that:


 * for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map {F_X} {x_n} \to 1$

at which point we will obtain:


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

Since $\sequence {x_n}_{n \mathop \in \N}$ is increasing:


 * the sequence $\sequence {\hointl {-\infty} {x_n} }_{n \mathop \in \N}$ is increasing.

We show that:


 * $\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \R$

Lemma
Let $P_X$ be the probability distribution of $X$.

So, from Measure of Limit of Increasing Sequence of Measurable Sets, we have:


 * $\ds \map {P_X} \R = \lim_{n \mathop \to \infty} \map {P_X} {\hointl {-\infty} {x_n} }$

So, we obtain:

Since $\sequence {x_n}_{n \mathop \in \N}$ was arbitrary, we have:


 * for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map {F_X} {x_n} \to 1$.

So, from Sequential Characterisation of Limit at Positive Infinity of Real Function: Corollary, we have:


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