Limit of Cumulative Distribution Function at Negative Infinity/Lemma

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Lemma

Let $\sequence {x_n}_{n \mathop \in \N}$ be a decreasing sequence with $x_n \to -\infty$.

Then:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$


Proof

{AimForCont} suppose that:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} \ne \O$

Let:

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

Then:

$x \in \hointl {-\infty} {x_n}$ for each $n$.

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

there exists $N \in \N$ such that $x_N < x$.

But then:

$x \not \in \hointl {-\infty} {x_N}$

contradicting that:

$x \in \hointl {-\infty} {x_n}$ for each $n$.

So we have reached a contradiction, and we have:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$

$\blacksquare$