Motivation behind the definition of complete metric space 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?

This answer only applies to the order version of completeness rather than the metric version, but I've found it quite a nice way to think about what completeness means intuitively: consider the real numbers. There the completeness property is what guarantees that the space is connected. The rationals can be split into disjoint non-empty open subsets, for example the set of all positive rationals whose squares are greater than two, and its complement, and the reason this works is because, roughly speaking, there is a "hole" in between the two sets which lets you pull them apart. In the reals this is not possible; there are always points at the ends of intervals, so whenever you partition the reals into two non-empty subsets, one of them will always fail to be open.
