Real Number

Definition

$\mathbb{R}$

Construction

Construction of the Real Numbers

Facts

Law of Trichotomy

Definition

$\forall a,b\in \mathbb{R},a>b\text{or}a=b\text{or}a<b$ Every Real Number is either positive, negative, or zero.

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When $s\in \mathbb{R}$ is an Upper Bound of $S$, the following are equivalence relations.

• $s=\sup (S)$
• $\forall ϵ>0,\exists x\in S,s-ϵ<x\le s$
• $\forall ϵ>0,S\cap (s-ϵ,s]\ne \varnothing$

Every closed interval in $\mathbb{R}$ is uncountable (by the Theorem)