Limit of Cumulative Distribution Function at Positive Infinity/Lemma

Lemma
Let $\sequence {x_n}_{n \mathop \in \N}$ be an increasing with $x_n \to \infty$.

Then:


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

Proof
Clearly we have:


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

So we only need to show that:


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

Let $x \in \R$.

From the definition of a sequence diverging to $\infty$:


 * there exists $N \in \N$ such that $x_N > x$.

So:


 * $x \in \hointl {-\infty} {x_N}$

giving:


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

So:


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

from the definition of subset.

So, we have:


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

from the definition of set equality.