![]() The use of Cauchy sequences is one of the two famous ways of defining the real numbers, that is, completing the rationals. The reals are the metric completion of the rationals. Metrically complete means that every Cauchy sequence made from the set converges to an element which is itself in the set. We have defined an extension to the rationals that is metrically complete-that extension of the rationals is the real numbers. Once we have done that, the payoff is enormous. We have to give the Cauchy sequences corresponding to rational numbers.We have to show how to add, subtract, multiply, and divide Cauchy sequences.So a real number is actually an "equivalence class" of Cauchy sequences, under a carefully defined equivalence. For example, the sequenceĬonsists only of rational numbers, but it converges to 2. The famous example of this is that a sequence of Rational Numbers might converge, but not to a rational number. A sequence of numbers in some set might converge to a number not in that set. ![]() ![]() The usual decimal notation can be translated to Cauchy sequences in a natural way. The definition above applies, as long as the Cauchy sequence (, ) is replaced with an arbitrary Cauchy net. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Generalizations to more abstract uniform spaces exist in the. There is an extremely profound aspect of convergent sequences. Cauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to nets (or equivalently filters). Cauchy sequences require the notion of distance so they can only be defined in a metric space. ![]() Cauchy sequences are named after the French mathematician Augustin Cauchy (1789-1857). This type of convergence has a far-reaching significance in mathematics. The reader should be familiar with the material in the Limit (mathematics) page.Ī Cauchy sequence (pronounced CO-she) is an infinite sequence that converges in a particular way. Educational level: this is a tertiary (university) resource. ![]()
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