7/7/2023 0 Comments Cauchy sequence definition![]() ![]() ![]() There is an extremely profound aspect of convergent sequences. A Cauchy sequence (pronounced CO-she) is an infinite sequence that converges in a particular way. Applying the formal definition of the limit of a sequence to prove that a sequence converges. Cauchy sequences are named after the French mathematician Augustin Cauchy (1789-1857). This type of convergence has a far-reaching significance in mathematics. Kelley, "General topology", Springer (1975)Ī.N. A Cauchy sequence (pronounced CO-she) is an infinite sequence that converges in a particular way. Simple exercise in verifying the de nitions. Proposition 3.1 If (X kk) is a normed vector space, then a sequence of points fX ig1 i1 Xis a Cauchy sequence i given any >0, there is an N2N so that i j>Nimplies kX i X jk< : Proof. Dugundji, "Topology", Allyn
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