Complete Metric Spaces

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Recall that a sequence in is called a Cauchy sequence if for all there exists an such that if then .

Consider any metric space . If is such that every Cauchy sequence converges in , then we give this metric space a special name.

Definition: Let be a metric space. Then is said to be Complete if every Cauchy sequence converges in .

In general, it is much easier to show that a metric space is not complete by finding a Cauchy sequence that does not converge in the space. For example, consider the set with the standard Euclidean metric defined for all . Then is a metric space. Now, consider the following sequence in :

We claim that is a Cauchy sequence. Let's prove this. Let , and consider:

Choose such that . Then if are such that then:

Hence for all we have that:

So is indeed a Cauchy sequence. However, it should be intuitively clear that , but ! Therefore does not converge in and hence is not a complete metric space.


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