# Show that (X,d) is complete

Suppose $$(X,d)$$ is a metric space and for all sequences $$\{x_n\}$$, if $$\sum d(x_n,x_{n+1})< \infty$$ then $$\{x_n\}$$ converges. Prove that $$(X,d)$$ is complete.

Question: I can show that if $$\sum d(x_n,x_{n+1})< \infty$$ then the sequence is Cauchy, but what if there is a Cauchy sequence that does not satisfy the condition $$\sum d(x_n,x_{n+1})< \infty$$, how can I prove that the space is complete?

An obvious counterexample is $$\mathbb R$$. However, if you meant '$$(X,d)$$ is complete' instead of '$$(X,d)$$ is compact' then the result is true. Given any Cauchy sequence $$\{x_n\}$$ find a subsequence $$\{x_{n_{k}}\}$$ such that $$\sum d(x_{n_{k}},x_{n_{k+1}})<\infty$$, conclude that $$\{x_{n_{k}}\}$$ converges. If a subseqeunce of a Cauchy sequence converges the whole sequence converges.