Show that sup(AUB)=max(sup(A),sup(B))

Kinetica

1. The problem statement, all variables and given/known data
Let A and B be nonempty bounded subsets of R. Show that sup(AUB)=max(sup(A),sup(B))

Or another version of this problem:
Show that if A and B are bounded subsets of R, then AUB is a bounded set. Show that sup(AUB)=sup(supA, supB)


3. The attempt at a solution

Hi! I have hard time visualizing this problem. Would you help me understand it visually? What should think of first when trying to solve this problem?

Does it matter at all if the subsets look like this or not?:

_____[ A ]______[ B ]_____

VS

_____{ A [ } B ]________


The solutions to this problem seems not to care about that. That's why I am confused.
For example, in the first version of the problem, they only care if sup A is equal, greater, or less than sup B. They somehow don't care if the subsets overlap or not.
And this bothers me a lot! :-(

Thank you in advance.

Dick

Why would it matter whether they overlap or not?? AUB contains both A and B whether they overlap or not. What's the definition of sup?