Definition:Complete Metric Space

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

Examples
The space $$\R$$ of real numbers is complete. (Real Number Line is Complete Metric Space.) 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.)