Question Number 215939 by mokys last updated on 21/Jan/25
$${for}\:{any}\:{natural}\:{numbers}\:{m},{n}\:{then}\:{m}={n}\:{or}\:{m}<{n}\:{or}\:{m}>{n}\:?\:{prove} \\ $$
Answered by MrGaster last updated on 22/Jan/25
$$\forall{m},{n}\in\mathbb{N} \\ $$$${m}={n}\vee{m}<{n}\vee{m}>{n} \\ $$$$\because\mathbb{N}\subset\mathbb{R}\wedge\mathbb{R}\:\mathrm{is}\:\mathrm{totally}\:\mathrm{ordered} \\ $$$$\mathrm{Assume}\:\mathrm{without}\:\mathrm{loss}\:\mathrm{of}\:\mathrm{generality}\:{m}\neq{n} \\ $$$${m}−{n}=\mathrm{0}\Rightarrow{m}={n} \\ $$$${m}−{n}>\mathrm{0}\Rightarrow{m}>{n} \\ $$$${m}−{n}<\mathrm{0}\Rightarrow{m}<{n} \\ $$$$\mathrm{By}\:\mathrm{trichotomy}\:\mathrm{property}\:\mathrm{of}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\therefore{m}={n}\vee{m}<{n}\vee{m}>\mathrm{0} \\ $$