# Determine if Sequence is Cauchy

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

• I think the partial decomposition should be $\frac{1}{n}-\frac{1}{n+1}$ which is telescoping!!\ – mathpadawan Apr 19 at 0:14
• You should use MathJax. Posts that do not use MathJax tend to get less attention. – parsiad Apr 19 at 0:14
• 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. – Ertxiem Apr 19 at 0:16

$$|a_n| <\frac 1 n$$ so $$|a_n-a_m|\leq |a_n|+|a_m| <\frac 1 n+\frac 1 m<\epsilon$$ if $$n,m >\frac 2 {\epsilon}$$.