Let A and B be bounded subsets of R. (a) Why does sup (AUB) exist? (b) Prove that sup (AUB) = max

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).