Suppose is a Cauchy sequence in a metric spare X, and some subsequence converges to a point . Show the full series converges to p.

Is this correct?

First prove convergence of . Find an integer N such that . Now set N so that for . Now replace with , then

Now prove that the sequence goes to p. Assume x and for . Pick an integer N such that This cannot be possible since it would violate the fact that is Cauchy, so