1. ## Supremum and infimum

Find the supremum and infimum of the following sets:

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

For this would you just start to sub numbers in:

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

And look at the limit as $\displaystyle n$ approaches infinity

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

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

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

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

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

I'm having some trouble with this one:

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

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

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