$(X,d)$ is a metric space with property that every closed and bounded set is compact.Now how can I show $X$ is complete? Can any one help me to give hints about it?
2026-04-25 04:46:45.1777092405
How to show that metric space $(X,d)$ is complete
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Hint: Start with a Cauchy sequence. Show you can put this in a compact set; now use the fact that metric and compact implies sequential compactness. This tells us that the Cauchy sequence has a convergent subsequence. Now you can show that the sequence itself converges to this limit point, showing that the original sequence you began with converges.
Since the Cauchy sequence was arbitrary, all Cauchy sequences must converge, so you are done.