I have been thinking about this for a while now.
Clearly if a sequence converges then also it will also have a convergent subsequence (take for example the whole sequence). However, I have been told the the opposite it not true. Could someone give an explicit example on a sequence which have a convergent subsequence but which do not converge?
Also we know that on a compact set, any bounded sequence has a convergent subsequence. Is there any criteria so that bounded sequences converges on compact set? Or more generally, is there a criteria so that if $x_n$ has a convergent subsequence then it will also converge?
Thanks.
Best Answer
You could take the sequence $a_n = (-1)^n$. What are the convergent subsequences?
It is necessary that a convergent sequence be a Cauchy sequence. Conversely, a Cauchy sequence with a converging subsequence converges.