Question Number 208103 by depressiveshrek last updated on 05/Jun/24
$$\mathrm{Find}\:\mathrm{inf}\left\{\frac{{m}}{{n}}\:\mid\:{m},\:{n}\:\in\:\mathbb{N},\:{m}<{n}−\mathrm{2}\right\} \\ $$
Answered by A5T last updated on 05/Jun/24
$${Since}\:{m},{n}\in\mathbb{N},\:\frac{{m}}{{n}}>\mathrm{0},{so}\:\mathrm{0}\:{is}\:{a}\:{lower}\:{bound}. \\ $$$$\frac{{m}}{{n}}\geqslant\frac{\mathrm{1}}{{n}}\:{and}\:\frac{\mathrm{1}}{{n}}\:{is}\:{in}\:{the}\:{set}. \\ $$$${For}\:{any}\:\epsilon>\mathrm{0},\:{one}\:{can}\:{always}\:{find}\:{n}\:{such}\:{that} \\ $$$$\frac{\mathrm{1}}{{n}}−\epsilon<\mathrm{0}.\:{Take}\:{n}>\frac{\mathrm{1}}{\epsilon}\:{which}\:{is}\:{possible}\:{by}\:{the} \\ $$$${Archimedean}\:{property},\:{hence}\:\mathrm{0}\:{is}\:{the}\:{infimum} \\ $$$${of}\:{the}\:{set}. \\ $$