Hello did a few exercises about supermum and infimum but im not sure if my solutions are correct. The following set are given:
(i) $\{n \text{ is element of the whole numbers } | n³ < 10\}$ Here was a little mistake. I think my answer is right now ?
(ii) {(3/n) +4 |n is element of the natural numbers (without 0)} Same translation mistake.
(iii) $\{ x\text{ element of real numbers } | x² < 5\}$
(iv) {(k-2)² < 10 | k { element of integers }
(v) { 3 + ((-1)^n)/(n) | n element of natural numbers (without 0) } Sorry i made a translation mistake.
My solutions are:
(i) $\sup = 2$
(ii) $\sup = 7$ ? inf = 4
(iii) $\sup = \sqrt{5}$
(iv) $\sup = 5, \inf = -1$
(v) $\sup = 3,5 \inf = 2$
My problem with exercises like these is : if you take (i) for instance. I know for this set n = 2. But is the supremum now 2 or 8 because 2³ = 8 ?
Question regarding (iii)
Im not exactly sure why it has to be $\sqrt 5$. ($\sqrt 5$)² < 5 is not true ?
One question regarding syntax:
Is this right: {n is element of the whole numbers | n³ < 10}; sup = 2
but
{n³ < 10 |n is element of the whole numbers}; sup = 8
After question was edited 1 min ago:
Your main problem is that you're not figuring out what the sets are before attempting to find their supremum or infimum.
(i) correct answer
(ii) correct answer; what about infimum?
(iii) correct answer
(iv) correct answer
(v) wrong answer; try $n = -1$?
As for your problems, well the thing is you have to first figure out what exactly the sets are before even attempting to figure out its supremum and infimum. That would answer your first problem. The second problem is answered by thinking about what is the supremum of the open interval $(0,1)$. It is $1$, even though $1$ is not an element in $(0,1)$.
For your query about syntax, "$\{ n^3 < 10 : n \in \mathbb{Z}_{\ne 0} \}$" is invalid syntax. You would have to write $\{ n^3 : n \in \mathbb{Z}_{\ne 0} \text{ and } n^3 < 10 \}$, and this set does have supremum $8$.