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Can someone please tell me how to determine if a sequence is Cauchy without using the limit. I'm supposed to use partial fraction decomposition

$a_n=\frac{1}{n(n+1)}$. When I did the partial decomposition I found $\frac{1}{n}+\frac{1}{n+1}$ but I don't know what to do from there.

Thank you

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    $\begingroup$ I think the partial decomposition should be $\frac{1}{n}-\frac{1}{n+1}$ which is telescoping!!\ $\endgroup$ – mathpadawan Apr 19 at 0:14
  • $\begingroup$ You should use MathJax. Posts that do not use MathJax tend to get less attention. $\endgroup$ – parsiad Apr 19 at 0:14
  • $\begingroup$ You seem to have two typos with the parenthesis, can you please edit your post and correct them? And don't forget to use MathJax to make things easier to read. $\endgroup$ – Ertxiem Apr 19 at 0:16
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$|a_n| <\frac 1 n$ so $|a_n-a_m|\leq |a_n|+|a_m| <\frac 1 n+\frac 1 m<\epsilon$ if $n,m >\frac 2 {\epsilon}$.

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