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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?

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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.

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