# The real numbers

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Algebraic structure The real numbers [ilmath]\mathbb{R} [/ilmath] TODO: Todo - is a field Main page: The real line [ilmath]\langle a,b\rangle:\eq a*b[/ilmath] - Euclidean inner-product on [ilmath]\mathbb{R}^1[/ilmath] [ilmath]\Vert x\Vert:\eq\sqrt{\langle x,x\rangle}\eq\vert x\vert[/ilmath] - Euclidean norm on [ilmath]\mathbb{R}^1[/ilmath] [ilmath]d(x,y):\eq\Vert x-y\Vert\eq \vert x-y\vert[/ilmath] - Absolute value - Euclidean metric on [ilmath]\mathbb{R}^1[/ilmath] topology induced by the metric [ilmath]d[/ilmath] Borel [ilmath]\sigma[/ilmath]-algebra of [ilmath]\mathbb{R} [/ilmath][Note 1] Lebesgue-measurable sets of [ilmath]\mathbb{R} [/ilmath] contains the Borel [ilmath]\sigma[/ilmath]-algebra

## Definition

### Cantor's construction of the real numbers

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

We further claim:

1. that the familiar operations of addition, multiplication and division are well defined and
2. 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]

TODO: Flesh out

## Properties

If [ilmath]S\subseteq\mathbb{R} [/ilmath] is a non-empty set of real numbers that has an upper bound then[2]:

• [ilmath]\text{Sup}(S)[/ilmath] (the supremum of [ilmath]S[/ilmath]) exists.

## Notes

1. This is just the Borel sigma-algebra on the real line (with its usual topology)