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November 7th, 2017, 05:52 AM   #1
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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?
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November 7th, 2017, 05:59 AM   #2
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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.
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November 7th, 2017, 02:47 PM   #3
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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.
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November 10th, 2017, 01:40 AM   #4
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Thanks, I proved it.

$$\frac{mn}{1+m+n}\ge \frac{1}{3} \rightarrow 3mn \ge m+n+1 \rightarrow m(n-1)+n(m-1)+mn-1 \ge 0$$

This is correct since $m,n \ge 1$.
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