# Definition:Complete Metric Space

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## Definition

### Definition 1

A metric space $M = \struct {A, d}$ is **complete** if and only if every Cauchy sequence is convergent.

### Definition 2

A metric space $M = \struct {A, d}$ is **complete** if and only if the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty.

## Equivalence of Definitions

These definitions are shown to be equivalent in Equivalence of Definitions of Complete Metric Space.

## Also see

- Results about
**complete metric spaces**can be found here.

## Sources

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- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous) ... (next): $3.12$ - 2020: James C. Robinson:
*Introduction to Functional Analysis*... (previous) ... (next) $4$: Complete Normed Spaces