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Concept
Cauchy sequence
Summary
In mathematics, a Cauchy sequence, 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 term. For instance, in the sequence of square roots of natural numbers:
a_n=\sqrt n,
the consecutive terms become arbitrarily close to each other – their differences
a_{n+1}-a_n = \sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}} < \frac{1}{2\sqrt n}
tend to zero as the index n grows. However, with growing values of n, the terms a_n become arbitrarily large. So, for any index n and distance d, there exists an index m big enough such that a_m - a_n > d.
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