(M,d) be a metric space with a property that every bounded sequence has a convergent subsequence $\Rightarrow$ M is complete
Proof To be complete we need to verify $\forall$ cauchysequence in $M$ converges in $M$.
first what we know is that all cauchy sequence is bounded sequence. then by the assumption, bounded sequence in M has a convergent subsequence in M.
Thus, M is complete.