Equivalence of Definitions of Complete Metric Space

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Theorem

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

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.


Proof

Definition 1 implies definition 2 by the Nested Sphere Theorem.


Let $M = \struct {A, d}$ be a complete metric space by definition 2.

Let $\sequence {x_n}$ 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 $n, m \in \N$:

$n, m \ge N_k \implies \map d {x_n, x_m} < \dfrac 1 {2^k}$

For each $k \in \N_{>0}$, let $C_k = \map { {B_{1/2^k} }^-} {x_{N_k} }$.


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Sources

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