A metric space $X$ is complete if every Cauchy sequence 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 $p \geq 1$