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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. 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. is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the completion of a given space, as explained below. Cauchy sequence A sequence in a metric space is called Cauchy if for every positive real number there is a positive integer such that for all positive integers Complete space A metric space is complete if any of the following equivalent conditions are satisfied: Every Cauchy sequence of points in has a limit that is also in Every Cauchy sequence in converges in (that is, to some point of ). Every decreasing sequence of non-empty closed subsets of with diameters tending to 0, has a non-empty intersection: if is closed and non-empty, for every and then there is a point common to all sets 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 and This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit then by solving necessarily yet no rational number has this property. However, considered as a sequence of real numbers, it does converge to the irrational number . The open interval , again with the absolute difference metric, is not complete either. The sequence defined by is Cauchy, but does not have a limit in the given space. However the closed interval is complete; for example the given sequence does have a limit in this interval, namely zero. The space R of real numbers and the space C of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space Rn, with the usual distance metric.

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Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

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