Cauchy sequence

mathematics

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A Cauchy sequence is a sequence of elements within a metric space where, for any small positive number, there's a point in the sequence such that the distance between any two terms beyond that point is less than that small number. Essentially, the terms of the sequence get arbitrarily close to each other as the sequence progresses.12

A metric space is a set with a distance function (a metric) that defines a non-negative distance between any two points, adhering to specific properties like symmetry and the triangle inequality. A complete metric space is one where every Cauchy sequence converges to a point within that space. The real numbers form a complete metric space, meaning all Cauchy sequences of real numbers converge to a real number.32

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analysis
Properties of the real numbers
…is said to be a Cauchy sequence if it behaves in this manner. Specifically, (a_n) is Cauchy if, for every ε > 0, there exists some N such that, whenever r, s > N, |a_r − a_s| < ε. Convergent sequences are always Cauchy, but is every Cauchy sequence convergent?…
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…is said to be a Cauchy sequence if it behaves in this manner. Specifically, (a_n) is Cauchy if, for every ε > 0, there exists some N such that, whenever r, s > N, |a_r − a_s| < ε. Convergent sequences are always Cauchy, but is every Cauchy sequence convergent?…
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metric space
…is the limit of a Cauchy sequence of rational numbers. In this sense, the real numbers form the completion of the rational numbers. The proof of this fact, given in 1914 by the German mathematician Felix Hausdorff, can be generalized to demonstrate that every metric space has such a completion.
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…is the limit of a Cauchy sequence of rational numbers. In this sense, the real numbers form the completion of the rational numbers. The proof of this fact, given in 1914 by the German mathematician Felix Hausdorff, can be generalized to demonstrate that every metric space has such a completion.
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