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**Let ## X ## be a random variable. Then ## \lim_{s \to \infty} P( |X| \geq s ) =0 ##**

*Theorem:***Proof from teacher assistant's notes:**We'll show first that ## \lim_{s \to \infty} P( X \geq s ) =0 ## and ## \lim_{s \to \infty} P( X \leq -s ) =0 ##:

Let ## (s_n)_{n=1}^\infty ## be a monotonically increasing sequence with ## \lim_{ n \to \infty } s_n = \infty ##. The sequences ## \{ X \geq s_n \}_{n=1}^\infty ## and ## \{ X \leq -s_n \}_{n=1}^\infty ## are decreasing sequences with zero intersection:

##\bigcap_{n=1}^{\infty}\left\{X \leq-s_n\right\} = \bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\} = \emptyset ##,

hence from continuity of probability:

##

\begin{aligned}

&0=\mathbb{P}(\emptyset)=\mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \geq s_n\right) \\

&0=\mathbb{P}(\emptyset)=\mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \leq-s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \leq-s_n\right)

\end{aligned}

##

Hence we'll deduce:

##

\lim _{s \rightarrow \infty} \mathbb{P}(|X| \geq s)=\lim _{s \rightarrow \infty}(\mathbb{P}(X \geq s)+\mathbb{P}(X \leq-s))=0

##

and we're finished.

*Questions:*1. I understand that the proof above is according to Heine's definition of limit, but if so I don't understand why we took ## (s_n)_{n=1}^\infty ## to be a monotonically increasing sequence and not an arbitrary sequence? ( we'd also like to prove for sequences that do go to infinity but are not necessarily monotonically increasing ).

2. Why does the equation ## \mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \geq s_n\right) ## hold? how did we go from the left side to the right side?

Thanks in advance for any help!