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 Cauchy sequence, DYK Fact #165
algebraic topology
Posted: 15:10 Friday 30 March 2007


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JaneFairfax
Posted: 20:38 Thursday 05 February 2009


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Did you know? Cauchy sequences of rational numbers can be added and multiplied term by term, and the results are also Cauchy sequences of rataional numbers. Indeed, under these operations, the set of all Cauchy sequences or rational numbers forms a commutative ring with zero element (0,0,0,…) and multiplicative identity (1,1,1,…). Moreover, the set of all Cuachy sequences of rational numbers which converge to 0 is an ideal of and the quotient ring is isomorphic to .

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algebraic topology
Posted: 10:41 Friday 06 February 2009


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Did you know?

The elements of are equivalence classes of the equivalence relation ~ defined on by .

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JaneFairfax
Posted: 14:01 Friday 06 February 2009


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Did you know? Cauchy sequences of rational numbers provide one of two principal methods of constructing the real numbers from the rational numbers. The other method, which I read about in A Survey of Modern Algebra by Garrett Birkhoff and Saunders MacLane a long time ago, is by means of Dedekind cuts. ThumbsUp.gif

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algebraic topology
Posted: 16:34 Friday 06 February 2009


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The construction of the reals from the rationals (whether by Cauchy sequences or by Dedekind cuts) involves the creation of a complete ordered field.

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JaneFairfax
Posted: 18:12 Friday 06 February 2009


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Did you know? Birkhoff & MacLane’s book also gives a proof of the result that any two complete ordered fields are isomorphic. cool.gif

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algebraic topology
Posted: 01:11 Wednesday 11 February 2009


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Did you know?

Cauchy sequences are named after the French mathematician Augustin Louis Cauchy.

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JaneFairfax
Posted: 16:30 Wednesday 11 February 2009


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Did you know? Dedekind cuts are named after the German mathematician Richard Dedekind. Waggish.gif

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