November 7th, 2017, 05:52 AM  #1 
Newbie Joined: Oct 2017 From: Italy Posts: 13 Thanks: 0  Strange sup & inf
Find the sup and inf of this set $$A={ \frac{mn}{1+m+n}}$$ With $m,n\in \mathbb{N}$. (Let be A a subset of $\mathbb{R}$). How to find them? I tried to change the variables, putting $a=n+m$ or $a=nm $ but it doesn't seem to work! Any idea? 
November 7th, 2017, 05:59 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,031 Thanks: 2342 Math Focus: Mainly analysis and algebra 
Think geometrically. What could $2(m+n)$ represent? And what then is $mn$? For any given value of $m+n$, how can you maximise/minimise $mn$? Last edited by v8archie; November 7th, 2017 at 06:07 AM. 
November 7th, 2017, 02:47 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,378 Thanks: 542 
There is no sup. For example, let m=n and let it get bigger, so A is approximately n (no bound). For inf, let m=n=1, then A=1/3. You need to verify that for larger values for n or m, A is larger. 
November 10th, 2017, 01:40 AM  #4 
Newbie Joined: Oct 2017 From: Italy Posts: 13 Thanks: 0 
Thanks, I proved it. $$\frac{mn}{1+m+n}\ge \frac{1}{3} \rightarrow 3mn \ge m+n+1 \rightarrow m(n1)+n(m1)+mn1 \ge 0$$ This is correct since $m,n \ge 1$. 

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inf, real analysis, sets, strange 
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