If your answer is hand written please post a clear copy so i

If your answer is hand written, please post a clear copy so it is readable.

Let (X, d) be a metric space, and (x_n)_n N a Cauchy sequence in (X, d). Show that, if (x_n)_n N has a subsequence (xk_n)_n N, with xk_n rightarrow x X (w.r.t. d), then also x_n rightarrow x.

Solution

Convergence:

A sequence (x_n) in X converges to x if for every neighborhood U of x there exists N N such that x_n U for all n N. In this case, x_n converges to x

Let (x_nk ) be a subsequence of (x_n). Take a neighborhood U of x. Since lim x_n = x, there is N N such that x_n U for all n N. From the definition of a subsequence, n_k k for all k N. So, n_k N for all k N. Thus x_nk U for all k N. Hence (x_nk ) converges to x as claimed.