Cauchy sequence [SEP] named after
Cauchy sequence
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:
the consecutive terms become arbitrarily close
The Cauchy–Davenport theorem named after Augustin Louis Cauchy and Harold Davenport asserts that for any prime "p" and nonempty subsets "A" and "B" of the prime order cyclic group Z/"p"Z we have the inequality
We may use this to deduce the Erdős–Ginzburg–Ziv theorem: given any sequence of 2"n"−1 elements in Z/"n", there are "n" elements that sums to zero modulo "n". (Here "n" does not need to be prime.)
A direct consequence of