Find inf{(m/n) ∣ m, n ∈ N, m<n−2}


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Question Number 208103 by depressiveshrek last updated on 05/Jun/24

$Find \inf {\frac{m}{n} ∣ m, n \in N, m < n - 2}$
	Answered by A5T last updated on 05/Jun/24

	$S i n c e m, n \in N, \frac{m}{n} > 0, s o 0 i s a l o w e r b o u n d .$ $\frac{m}{n} ⩾ \frac{1}{n} a n d \frac{1}{n} i s i n t h e s e t .$ $F o r a n y ϵ > 0, o n e c a n a l w a y s f i n d n s u c h t h a t$ $\frac{1}{n} - ϵ < 0 . T a k e n > \frac{1}{ϵ} w h i c h i s p o s s i b l e b y t h e$ $A r c h i m e d e a n p r o p e r t y, h e n c e 0 i s t h e i n f i m u m$ $o f t h e s e t .$
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