Cauchy sequence

Noun

(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every 𝜀>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<𝜀.

Japanese Meaning

数学解析における概念で、距離空間内の点列（系列）が、任意の正の実数εに対して、ある自然数N以降の全ての項同士の距離がε未満となる性質を持つものである。

What is this buttons?

In a complete metric space, every Cauchy sequence converges to a limit in that space.

Google Translation / DeepL Translation

完備な距離空間では、任意のコーシー列はその空間で収束し、極限を持ちます。

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Sense（1）

(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every 𝜀>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<𝜀.

However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.

Google Translation / DeepL Translation

Details

Cauchy sequences

（ plural ）

Details

Cauchy sequence

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