hm... every converging sequence is Cauchy sequence

Th : "every compact metric space is complete .... "

if X is compact metric space and let be Cauchy sequence in X. because of compactness there is sub sequence of our sequence.. and sub sequence is converging now we have

now first on the right side can be made very small because it is Cauchy , and second to because of convergence of the sub sequence so we have

meaning sequence is converging and space X is complete