How do I find the supremum and infimum of this set

Question

$$A=\left\{\dfrac{mn}{1+m+n} \mid m,n \in \mathbb N \right\}$$

I'm relatively new to this whole infimum and supremum proving. I've tried for a long time to prove the infimum of this set (which I believe to be 1/3). Can someone please help me and provide a proof for that?

Infimum $inf \ A = 0$ if you consider the natural numbers with $0$, if not its $\frac{1}{3}$, since this sequence is increasing in both $m$ and $n$. — BIS HD, Oct 26 '13 at 18:09
Note that for $m=n^2-1$, one has $\frac{mn}{1+m+n}=n-1$. This shows ${\sf sup}(A)=\infty$ — Ewan Delanoy, Oct 26 '13 at 18:11

score 1 · Accepted Answer · answered Oct 26 '13 at 18:14

1

Hint: $$0 < \dfrac1{mn} + \dfrac1{m} + \dfrac1{n} \le 3, \qquad m,n \in \mathbb{N}.$$

answered Oct 26 '13 at 18:14

njguliyev

14,473
1
26
43

As long as you consider the natural numbers without 0. – BIS HD Oct 26 '13 at 18:15
@BISHD, I know only one definition of $\mathbb{N}$: $1,2,3, \ldots$ – njguliyev Oct 26 '13 at 18:19
In fact there is no universal agreement, see for example wikipedia – BIS HD Oct 26 '13 at 18:22

score 1 · Answer 2 · answered Oct 26 '13 at 18:15

I think it might be helpful to rewrite it slightly:

$$ \frac{mn}{1+m+n}=\frac{n}{\frac{1}{m}+1+\frac{n}{m}} $$

For fixed $n$, as $m$ increases this increases as well. Same goes for the reversed case. Hence, $\sup A$ should be $\infty$. For as small a value as possible, $m$ should then be as small as possible since this means that the denominator is as large as possible. So $m=1$. What's left is

$$ \frac{n}{2+n}=\frac{1}{\frac{2}{n}+1} $$ which obviously is as small as possible when the denominator is as large as possible -- i.e. when $n=1$. So $\inf=1/3$, which is achieved when $n=m=1$.

score 0 · Answer 3 · answered Jul 28 '20 at 05:31

0

Hint : Fix one variable, say m= 1 and take limit as n goes to infinity. Do that with other variable. Now put $m = n = t $ and take $t \to \infty$ , you get supremum to be infinity.

answered Jul 28 '20 at 05:31

Jessica Griffin

222
1
10

How do I find the supremum and infimum of this set

3 Answers3