# cauchy subsequence question

• Oct 21st 2011, 02:39 PM
wopashui
cauchy subsequence question
A sequence $(x_n)$ has a Cauchy subsequence if and only if it has a subsequence $(x_n_k)$ for which $d(x_n_k,x_n_{k+1}) < 2^{-k}$
• Oct 21st 2011, 05:25 PM
Drexel28
Re: cauchy subsequence question
Quote:

Originally Posted by wopashui
A sequence $(x_n)$ has a Cauchy subsequence if and only if it has a subsequence $(x_n_k)$ for which $d(x_n_k,x_n_{k+1}) < 2^{-k}$

what have you tried, once again? This one is slightly tricky, the key thing to note is that $\displaystyle \sum2^{-k}=1$. So, use the triangle inequality, etc.