Limit of Cumulative Distribution Function at Positive Infinity/Lemma

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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.

$\blacksquare$