Cauchy sequence

In mathematics, a Cauchy sequence (French pronunciation: ​[koʃi]; English pronunciation: /ˈkoʊʃiː/ KOH-shee), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

It is not sufficient for each term to become arbitrarily close to the preceding term. For instance, in the harmonic series ${\textstyle \sum {\frac {1}{n}}}$ a difference between consecutive terms decreases as ${\displaystyle {\tfrac {1}{n}}}$, however the series does not converge. Rather, it is required that all terms get arbitrarily close to each other, starting from some point. More formally, for any given ${\displaystyle \varepsilon >0}$ (which means: arbitrarily small) there exists an N such that for any pair m,n > N, we have ${\displaystyle |a_{m}-a_{n}|<\varepsilon }$ (whereas ${\displaystyle |a_{n+1}-a_{n}|<\varepsilon }$ is not sufficient).

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