Question

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

Answer 1

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).