# Thread: Supremum and infimum

1. ## Supremum and infimum

Find the supremum and infimum of the following sets:

1) $\left\{\frac{n}{1+n^2} : n=1,2...\right\}$

For this would you just start to sub numbers in:

$\{\frac{1}{2},\frac{2}{5}\}$

And look at the limit as $n$ approaches infinity

$\displaystyle\lim_{n\to\infty}\frac{n}{1+n^2}=0$

So therefore the supremum is $\frac{1}{2}$ and the infimum is $0$

2) $\left\{\frac{n}{1+n^2} : n\in\mathbb{Z}\right\}$

Once again would you just substitute numbers to see what is happening so the supremum and infimum would be:

$-\frac{1}{2}$ and $\frac{1}{2}$ respectively

I'm having some trouble with this one:

3) $\{x\in\mathbb{Q} : x^2<2\}$

2. Originally Posted by acevipa
Find the supremum and infimum of the following sets:
I'm having some trouble with this one:
3) $\{x\in\mathbb{Q} : x^2<2\}$
Consider $\pm\sqrt2$

3. $\sqrt{2}$ and $-\sqrt{2}$ are not in $\mathbb{Q}$, of course, but the supremum and infimum of a set don't have to be in the set.