# Equivalence of Definitions of Complete Metric Space

## Theorem

The following definitions of the concept of **Complete Metric Space** are equivalent:

### Definition 1

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

### Definition 2

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

## Proof

Definition 1 implies definition 2 by the Nested Sphere Theorem.

Let $M = \left({A, d}\right)$ be a complete metric space by definition 2.

Let $\left\langle{x_n}\right\rangle$ be a Cauchy sequence in $M$.

By the definition of a Cauchy sequence, for each $k \in \N_{>0}$, there is an $N_k \in \N_{>0}$ such that for all $n, m \in \N$:

- $n, m \ge N_k \implies d \left({x_n, x_m}\right) < \dfrac 1 {2^k}$

For each $k \in \N_{>0}$, let $C_k = B^-_{1/2^k} \left({x_{N_k}}\right)$.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 5$: Complete Metric Spaces