Answer:
a) sup(AUB) exist because A and B are bounded.
The definition of sup(A)={x∈A/y∈A,y≤x}
If x=sup(A),x∈A ⇒ x∈(A∪B)
If z=sup(B), z∈B ⇒ z∈(A∪B)
b)The value of sup(A∪B)=max(sup(B),sup(A))
proof
x∈A∪B⇒x∈A ∨ x∈B⇒x≤sup(A) ∨ x≤sup(B) then x≤max{sup(A),sup(B)}
sup(A)≤sup(A∪B) and
sup(B)≤sup(A∪B)
By definition of max:
max{sup(A),sup(B)}≤sup(A∪B).