A Cauchy sequence is a sequence in a metric space where the distance between successive terms becomes arbitrarily small as the sequence progresses, indicating convergence to a limit within a complete space. This concept is crucial in analysis as it provides a criterion for convergence that does not depend on knowing the limit beforehand, making it especially useful in spaces where limits may not be easily identifiable.

Cauchy Sequence

Cauchy sequence

A metric space is a set equipped with a function called a metric that defines a distance between any two elements in the set, allowing for the generalization of geometrical concepts such as convergence and continuity. This structure is fundamental in analysis and topology, providing a framework for discussing the properties of spaces in a rigorous mathematical manner.

metric space

distance between successive terms

Arbitrarily small distance

convergence to a limit

A complete space in mathematics is a metric space where every Cauchy sequence converges to a limit within the space. This property is fundamental in analysis as it ensures the space is closed under the operation of taking limits, making it robust for various mathematical operations and proofs.

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