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fishturtle1

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## Homework Statement

Prove the ##\limsup \vert s_n \vert = 0## iff ##\lim s_n = 0##.

## Homework Equations

##\limsup s_n = \lim_{N\rightarrow \infty} \sup \lbrace s_n : n > N \rbrace = \sup \text{S}##

##\liminf s_n = \lim_{N\rightarrow \infty} \inf \lbrace s_n : n > N \rbrace = \inf \text{S}##

Definition of limit: ##\lim s_n = L## iff For all ##\varepsilon > 0## there exists ##N(\varepsilon)## such that ##n > N(\varepsilon)## implies ##\vert s_n - L \vert < \varepsilon##.

## The Attempt at a Solution

##(\leftarrow)## Suppose ##\lim s_n = 0##. Then for all ##\varepsilon > 0## there exists ##N(\varepsilon)## such that ##\vert s_n - 0 \vert = \vert s_n \vert < \varepsilon##. So ##\vert \vert s_n \vert \vert < \varepsilon##. So ##\lim \vert s_n \vert = 0##. Since ##\lim \vert s_n \vert = 0 \epsilon \mathbb{R}##, we can say ##\limsup \vert s_n \vert = 0##.

##(\rightarrow)## Suppose ##\limsup \vert s_n \vert = 0##.