Completion (metric space)

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. √2 is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to "fill all the holes", leading to the completion of a given space, as explained below.

The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. Consider for instance the sequence defined by ${\displaystyle \scriptstyle x_{1}\;=\;1}$ and ${\displaystyle \scriptstyle x_{n+1}\;=\;{\frac {x_{n}}{2}}\,+\,{\frac {1}{x_{n}}}}$. This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then necessarily x² = 2, yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the irrational number √2.

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