Convergent Subsequence

Cauchy Sequence - Completeness - Other Properties
... Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ε > 0, beyond some fixed point, every term of sequence is within distance ε/2 of s, so any two ... In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given any real number r > 0, beyond ... that every bounded sequence of real numbers has a convergent subsequence ...