5

Let $A_1, A_2, A_3,\dots$ be a collection of nonempty sets, each of which is bounded above.

$(a)$ Find a formula for $\sup(A_1 \cup A_2)$. Extend this to supremum of a collection of $n$ sets $A_1, A_2, \dots, A_k$.

For $(a)$ I want to say that it's just the largest of the supremums, but I'm not sure how to show or prove that.

$(b)$ Consider the supremum of an infinite number of sets. Does the formula in $(a)$ extend to the infinite case?

For $(b)$ is it possible to have a supremum of an infinite number of sets as long as they're all bounded above?

Community
  • 1
user268486
  • 221
  • Welcome to Mathematics Stack Exchange. –  Sep 06 '15 at 19:01
  • Are these subsets of the real line $\Bbb R$? – Berci Sep 06 '15 at 19:03
  • Yes they are Berci – user268486 Sep 06 '15 at 19:05
  • I think $\sup(A_1 \cup A_2)=\sup({\sup(A_1),\sup(A_2)})$. – SMA.D Sep 06 '15 at 19:07
  • For part (a) : https://math.stackexchange.com/questions/921975/let-a-b-subset-mathbbr-show-that-supa-cup-b-max-sup-a-sup-b?rq=1 – Arnaud D. Oct 06 '17 at 12:16
  • for part (b): No, it does not necessarily hold. for counter examples consider $A_n=[0,n]$ or $B_n=[0,1-\frac{1}{n}]$ . – abhishek Apr 20 '21 at 15:10
  • $\cup_{n=1}^{\infty}A_n=[0,\infty]$ is not bounded above, and hence does not have a $\sup$. $\cup_{n=1}^{\infty}B_n=[0,1)$ whose $\sup$ is $1$, and $\sup B_n<1, \forall n$ – abhishek Apr 20 '21 at 15:13
  • solution copied from here---->https://users.math.msu.edu/users/chamatth/teaching/320spring19/hw1_sol.pdf – abhishek Apr 20 '21 at 15:24
  • Could we say $\operatorname{sup}\bigcup_{k=1}^{\infty}{A_k}=\operatorname{sup}\bigcup_{k=1}^{\infty}{{\operatorname{sup}{A_k}}}$ and would that extend to the infinite case for at least bounded unions? – lukejanicke Jun 30 '21 at 00:55

1 Answers1

4

a) Yes, we have $\ \sup(A_1\cup A_2)=\max(\sup A_1,\,\sup A_2)$.
To prove this, just use the definition of $\sup$ (e.g. for a subset $U$ and an element $v$ we have $U\le v\iff \sup U\le v\ $ where $U\le v$ wants to mean $u\le v$ for all $u\in U$.)

b) Well, the supremum can also be $+\infty$, and yes, it is possible to achieve, for a simplest example take $A_n:=\{n\}$. Each of these sets is of course bounded, but their union is not.
Can you find the formula for $\sup(A_1\cup A_2\cup\dots)$?

Berci
  • 90,745
  • That's helpful! Thank you! So is what you're saying for (b) that although the sets are bounded, the union of the sets is not and therefore there is no formula for the supremum? – user268486 Sep 06 '15 at 21:28
  • Yes, there is formula (if we also allow $+\infty$, but why not?). And that formula is basically on this page. – Berci Sep 07 '15 at 20:10
  • 1
    What if ∞ was not allowed? Then we wouldn't be able to find a supremum of an infinite amount of sets, correct? – mathmajor Sep 11 '18 at 00:16