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What is motivation behind the definition of a complete metric space?

Intuitively,a complete metric is complete if they are no points missing from it.

How does the definition of completeness (in terms of convergence of cauchy sequences) show that?

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    Completeness of metric spaces is a generalization of the completeness of the real numbers. It is one of the fundamental properties of the real numbers that all Cauchy sequences converge. (This isn't true, for example, in the rational numbers.) It is equivalent to whatever other form of completeness you may be aware of, e.g. that every bounded set has a least upper bound.2010-10-15

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