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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?

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  • $\begingroup$ Welcome to Mathematics Stack Exchange. $\endgroup$
    – user249332
    Sep 6, 2015 at 19:01
  • $\begingroup$ Are these subsets of the real line $\Bbb R$? $\endgroup$
    – Berci
    Sep 6, 2015 at 19:03
  • $\begingroup$ Yes they are Berci $\endgroup$
    – user268486
    Sep 6, 2015 at 19:05
  • $\begingroup$ I think $\sup(A_1 \cup A_2)=\sup(\{\sup(A_1),\sup(A_2)\})$. $\endgroup$
    – SMA.D
    Sep 6, 2015 at 19:07
  • $\begingroup$ For part (a) : math.stackexchange.com/questions/921975/… $\endgroup$
    – Arnaud D.
    Oct 6, 2017 at 12:16

1 Answer 1

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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 $\sup(A_1\cup A_2\cup\dots)$?

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  • $\begingroup$ 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? $\endgroup$
    – user268486
    Sep 6, 2015 at 21:28
  • $\begingroup$ Yes, there is formula (if we also allow $+\infty$, but why not?). And that formula is basically on this page. $\endgroup$
    – Berci
    Sep 7, 2015 at 20:10
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    $\begingroup$ What if ∞ was not allowed? Then we wouldn't be able to find a supremum of an infinite amount of sets, correct? $\endgroup$
    – mathmajor
    Sep 11, 2018 at 0:16

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