Complete metric space

A metric space in which every Cauchy sequence converges to a point in the space. Completeness means that limits of internally approximating sequences are not missing.

Cauchy sequence: A sequence whose terms become arbitrarily close to one another as their indices grow. Completeness is defined by requiring every such sequence to converge within the space.
Banach fixed-point theorem: A theorem stating that a contraction on a complete metric space has a unique fixed point. Completeness ensures the successive approximations converge to the fixed point.
Rational numbers: The numbers expressible as ratios of integers with nonzero denominator. With the usual distance, rational numbers are not complete: rational Cauchy sequences can converge to irrationals.
Uniform limit theorem: A theorem that a uniform limit of continuous functions is continuous. Completeness of the real or complex numbers supports the pointwise limits used in its proof.
Convergent sequence: A sequence whose terms approach a limit in its ambient space. Completeness requires Cauchy sequences to have this kind of limit inside the space.
Banach space: A complete normed vector space. Banach spaces use metric completeness to support analysis of infinite-dimensional vectors.
Real numbers: The complete ordered field extending the rational numbers with limits for bounded increasing sequences. The usual metric makes the real numbers a central example of a complete space.
Cauchy criterion: A condition for sequence convergence expressed through terms becoming arbitrarily close to one another. In complete spaces, satisfying this criterion is enough to guarantee a limit in the space.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Completeness adds a condition on sequences to this underlying distance structure.
Picard–Lindelöf theorem: An existence-and-uniqueness theorem for solutions of ordinary differential equations with locally Lipschitz right-hand sides. Its proof constructs solutions through a contraction argument on a complete function space.

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