Difference between revisions of "The real numbers"

	The real numbers
	[ilmath]\mathbb{R} [/ilmath]

Revision as of 11:49, 2 June 2016

Stub grade: C

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Once cleaned up and fleshed out, demote to D

The set of real numbers, [ilmath]\mathbb{R} [/ilmath], is the quotient space, [ilmath]\mathscr{C}/\sim[/ilmath] where:^[1]

[ilmath]\mathscr{C} [/ilmath] - the set of all Cauchy sequences in [ilmath]\mathbb{Q} [/ilmath] - the quotients
[ilmath]\sim[/ilmath] - the usual equivalence of Cauchy sequences

We further claim:

that the familiar operations of addition, multiplication and division are well defined and
by associating [ilmath]x\in\mathcal{Q} [/ilmath] with the sequence [ilmath] ({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q} [/ilmath] where [ilmath]\forall n\in\mathbb{N}[x_n:=x][/ilmath] we can embed [ilmath]\mathbb{Q} [/ilmath] in [ilmath]\mathbb{R}:=\mathscr{C}/\sim[/ilmath]