Let $A_1,A_2,A_3, ...$ be a collection of nonempty sets in $\mathbb R$, each of which is bounded above. The formula for $\sup \bigcup^{n}_{k=1}A_k = \sup \{\sup A_k | k = 1, 2, ...\}$. My question is: why it can't be $max\{\sup A_k | k = 1, 2, ...\}$? If all supremums exist, then taking the greatest of them would make it valid upper bound for that set, and I can't really imagine the set for which it won't be least upper bound. I've seen other answers to this exact question from Understanding Analysis, but I haven't found one to explain to me difference between "supremum of supremums" and "maximum of supermums" situation. Is there any?
$\begingroup$
$\endgroup$
3
asked Aug 9, 2022 at 15:18
-
$\begingroup$ Yes, taking the supremum on a finite set is the same as taking the maximum. $\endgroup$– FalconAug 9, 2022 at 15:22
-
$\begingroup$ Are you talking about a finite collection of sets only? $\endgroup$– pjq42Aug 9, 2022 at 15:23
-
$\begingroup$ Why are you assuming that the set $\left\{\sup A_k\,\middle|\,k\in\{1,2,\ldots\}\right\}$ has a maximum? It is not true in general. $\endgroup$– José Carlos SantosAug 9, 2022 at 15:42
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
Well the supremum and maximum are the same if you're considering a finite set. However say you're dealing with an infinite collection. Then you can't necessary guarantee a maximum exists. Take, for example, the sequence $\{A_j\}_{j\in\mathbb{N}}$ of intervals
$$A_j=[j,j+1).$$
Clearly each of these are bounded, and
$$\sup A_j=j+1,$$
however
$$\bigcup_{j\in\mathbb{N}}A_j=[0,\infty),$$
which is unbounded, and does not have a maximum, and indeed
$$\sup_{j\in\mathbb{N}}\sup A_j=\sup_{j\in\mathbb{N}}(j+1)=\infty.$$
$\begingroup$
$\endgroup$
It seems to me that as long as you are taking the union of a finite number of sets, then the supremum of suprema and the maximum of suprema are really the same thing.
If all sets are bounded, their suprema are finite. Thus, the set of all suprema becomes a finite set of real numbers which is again bounded. Therefore, its supremum (i.e. the supremum of suprema) should be finite and it should be just the maximum of the set as you say.
However, there could be an issue if you start taking the union of an infinite number of sets. Then, despite the fact that all suprema are finie, the set of suprema is an infinite set which is not necessarily bounded, and thus the supremum of suprema might not be finite and it might not be equal to the maximum which on the other hand should be finite.
