$A_1, A_2, A_3\dots $ are a collection of nonempty sets, each bounded above.
I'm asked to find a formula for $\sup(A_1\cup A_2)$ and then to extend this to $\bigcup^{n}_{k=1}A_k$.
For the supremum of $A_1\cup A_2$ I have shown that it is $\max\{a_1, a_2\}$ when $a_1$ and $a_2$ are the suprema of $A_1$ and $A_2$ which we know they must have because of the completeness axiom.
I'm having trouble extending it to $\bigcup^{n}_{k=1} A_k$. I am told to do so by induction. I proved it for the case $n=2$ in the first step, and I think my induction hypothesis is to suppose that it is true for $\bigcup^{n}_{k=1}A_k$ and then use this to show it is also true for $\bigcup^{n+1}_{k=1}A_k$. This is where I'm stuck.
$\sup(A \cup B \cup C) = \sup( (A \cup B) \cup C ) = \max( \sup(A \cup B) , \sup(C) )$ [by the 2-set case]
$\ = \max( \max( \sup(A) , \sup(B) ) , \sup(C) )$ [again] $ = \max( \sup(A) , \sup(B) , \sup(C) )$.
I'm sure you can see how to extend this to arbitrary finite number of sets!