Cauchy sequence, DYK Fact #165
 algebraic topology Posted: 15:10 Friday 30 March 2007 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007
 JaneFairfax Posted: 20:38 Thursday 05 February 2009 The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 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 $\mathcal{C}$ 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 $\mathcal{N}$ of all Cuachy sequences of rational numbers which converge to 0 is an ideal of $\mathcal{C}$ and the quotient ring $\mathcal{C}/\mathcal{N}$ is isomorphic to $\mathbb{R}$.
 algebraic topology Posted: 10:41 Friday 06 February 2009 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? The elements of $\mathcal{C}/\mathcal{N}$ are equivalence classes of the equivalence relation ~ defined on $\mathcal{C}$ by $\left(a_n\right)_{n=1}^{\infty}\sim\left(b_n\right)_{n=1}^{\infty}\;\Leftrightarrow\;\lim_{n\to\infty}(a_n-b_n)=0$.
 JaneFairfax Posted: 14:01 Friday 06 February 2009 The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 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.
 algebraic topology Posted: 16:34 Friday 06 February 2009 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? The construction of the reals from the rationals (whether by Cauchy sequences or by Dedekind cuts) involves the creation of a complete ordered field.
 JaneFairfax Posted: 18:12 Friday 06 February 2009 The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 Did you know? Birkhoff & MacLane’s book also gives a proof of the result that any two complete ordered fields are isomorphic.
 algebraic topology Posted: 01:11 Wednesday 11 February 2009 Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 Did you know? Cauchy sequences are named after the French mathematician Augustin Louis Cauchy.
 JaneFairfax Posted: 16:30 Wednesday 11 February 2009 The Enlightenment Group: Moderators Posts: 670 Member No.: 20 Joined: 03 Mar 2007 Did you know? Dedekind cuts are named after the German mathematician Richard Dedekind.
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