# complete

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

Examples:

• The space $\mathbb{Q}$ of rational numbers is not complete: the sequence $3$, $3.1$, $3.14$, $3.141$, $3.1415$, $3.14159$, $3.141592\ldots$ consisting of finite decimals converging to $\pi\in\mathbb{R}$ is a Cauchy sequence in $\mathbb{Q}$ that does not converge in $\mathbb{Q}$.
• The space $\mathbb{R}$ 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 $p\geq 1$.