View source for Cantor's construction of the real numbers/Definition

<noinclude>
{{Stub page|grade=A|msg=Somewhat badly written, demote to B once cleaned up}}
==Definition==
</noinclude>
The set of [[real numbers]], {{M|\mathbb{R} }}, is the [[quotient space (equivalence relation)|quotient space]], {{M|\mathscr{C}/\sim}} where:{{rAPIKM}}
* {{M|\mathscr{C} }} - the [[set]] of all [[Cauchy sequence|Cauchy sequences]] in {{M|\mathbb{Q} }} - [[the quotients]]
* {{M|\sim}} - the usual [[equivalence of Cauchy sequences]]
We further claim:
# that the familiar operations of addition, multiplication and division are well defined and
# by associating {{M|x\in\mathcal{Q} }} with the sequence {{MSeq|x_n|in=\mathbb{Q} }} where {{M|1=\forall n\in\mathbb{N}[x_n:=x]}} we can [[embedding|embed]] {{M|\mathbb{Q} }} in {{M|1=\mathbb{R}:=\mathscr{C}/\sim }}
<noinclude>
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Real Analysis}}
</noinclude>