Definition:Complete Metric Space

Definition
A metric space $\left({X, d}\right)$ is complete if every Cauchy sequence is convergent.

Alternative Definition
A metric space $\left({X, d}\right)$ is complete iff the intersection of every nested sequence of closed balls whose radii tend to zero is non-empty.

Equivalence of Definitions

 * These two definitions are logically equivalent.

Examples
The space $\R$ of real numbers is complete. More generally, Euclidean space $\R^n$ is complete.

On the other hand, the rational numbers $\Q$ do not form a complete metric space.

Indeed, any sequence of rational numbers that converges to an irrational number (in the metric space $\R$) is a Cauchy sequence that does not converge in $\Q$.

An example of such a sequence is given by
 * $\displaystyle a_n := \frac {f_{n+1}} {f_n}$

where $\left\langle{f_n}\right\rangle$ is the sequence of Fibonacci numbers.

This sequence converges to the golden mean:
 * $\displaystyle \lim_{n \to \infty} a_n = \phi := \frac{1 + \sqrt 5} 2$

which is irrational, as the square root of any prime is irrational:

Rationality of $\phi$ would make $\sqrt 5$ rational as well, a contradiction.