(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]x_{n}[/tex] be a Cauchy sequence. Suppose that for every [tex]\epsilon>0[/tex] there is [tex] n > \frac{1}{\epsilon}[/tex] such that [tex]|x_{n}| < \epsilon[/tex]. Prove that [tex]x_{n} \rightarrow 0[/tex].

2. Relevant equations

3. The attempt at a solution

My problem with the question is I do not understand it.

if,

[tex]|x_{n}| < \epsilon[/tex] when [tex] n > \frac{1}{\epsilon}[/tex] ;

Doesn't that mean that [tex]Lim_{n\rightarrow infinity} x_{n} = 0 ?[/tex]

In which case [tex]x_{n} \rightarrow 0[/tex] because if the limit as n goes to infinity is zero, then the terms can be arbitrarily brought close to zero as n gets large enough.

What exactly is the point of me knowing that [tex]x_{n}[/tex] is a Cauchy sequence ?

It doesn't seem like I even need to know that it is a Cauchy sequence. After-all, every convergent sequence is a cauchy sequence.

Am I missing something ?

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# Homework Help: Cauchy sequence; I need some help

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