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

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

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]

Axiomatic construction of the real numbers

Axiomatic construction of the real numbers/Definition

[ilmath]\mathbb{R} [/ilmath] is an example of:


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

References

  1. Analysis - Part 1: Elements - Krzysztof Maurin
  2. Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha