Definition:Complete Metric Space

Definition
A metric space $$(X,d)$$ is called complete if every Cauchy sequence is convergent.

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
 * $$a_n := \frac{f_{n+1}}{f_n}$$

where $$(f_n)$$ is the sequence of Fibonacci numbers.

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

which is irrational.