Fundamental sequence - Kihonretsu (English spelling)

Japanese: 基本列 - きほんれつ（英語表記）fundamental sequence

A sequence { a _n } where | a _m - a _n | approaches 0 infinitesimally as m and n become infinitesimally large is called a fundamental sequence or a Cauchy sequence. A sequence that converges is a fundamental sequence. When { a _i } and { b _i } are both fundamental sequences, { a _i + bi }, { a _i - _bi _} , and { a _i bi _} are all fundamental sequences. Also, if { bi _} does not converge to 0 and _all bi are not 0, then { a _i / bi _} is also a fundamental sequence. A fundamental sequence of real numbers always has a limit value. This property is called the completeness of the real numbers.

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

m,nが限りなく大きくなるとき，｜a_m－a_n｜が0に限りなく近づいていく数列｛a_n｝を基本列，またはコーシー列という。収束する数列は基本列である。｛a_i｝，｛b_i｝がともに基本列であるとき，｛a_i＋b_i｝，｛a_i－b_i｝，｛a_ib_i｝はいずれも基本列になる。また，｛b_i｝が0に収束せず，すべてのb_iが0でないならば，｛a_i/b_i｝も基本列になる。実数の基本列はつねに極限値をもつ。この性質を実数の完備性という。

出典　株式会社平凡社世界大百科事典第２版について　情報

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