complete

A metric space $X$ is complete if every Cauchy sequence (http://planetmath.org/CauchySequence) in $X$ is a convergent sequence.

Examples:

Cauchy sequence

• The space $\mathbb{Q}$ of rational numbers is not complete: the sequence $3$, $3.1$, $3.14$, $3.141$, $3.1415$, $3.14159$, $3.141592\mathrm{\dots }$ consisting of finite decimals converging to $\pi \in ℝ$ is a Cauchy sequence in $\mathbb{Q}$ that does not converge in $\mathbb{Q}$.

• The space $ℝ$ of real numbers is complete, as it is the completion of $\mathbb{Q}$ with respect to the standard metric (other completions, such as the $p$-adic numbers, are also possible). More generally, the completion of any metric space is a complete metric space.

• Every Banach space is complete. For example, the $L^p$–space of p-integrable functions is a complete metric space if $p\ge 1$.