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