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# Cauchy sequence

Let $$X$$ be a metric space. Then a sequence $$(x_1,x_2,x_3,\ldots)$$ of elements of $$X$$ is said to be Cauchy if given any $$\varepsilon > 0$$ there is a natural number $$N$$ such that the distance between $$x_i$$ and $$x_j$$ is less than $$\varepsilon$$ whenever $$i$$ and $$j$$ are greater than $$N$$. In effect, successive elements of the sequence eventually become arbitrarily close together.

A metric space is said to be complete if and only if every Cauchy sequence is also a convergent sequence.

Citation Info

• [MLA] “Cauchy sequence.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 19 Mar 2013. Web. 19 Mar 2013. <http://platonicrealms.com/>
• [APA] Cauchy sequence (19 Mar 2013). Retrieved 19 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/Cauchy-sequence/