please explain the solution and what it is wrong with my conception. FOLLOW the COMMENT PLEASE...

please explain the solution and what it is wrong with my conception. FOLLOW the COMMENT PLEASE

Question:

Let A and B are nonempty set bounded subset of R, and let A+B be the set of all sums a+b where a belngs to A and b belongs to B

Prove Sup(A+B)=Sup(A)+Sup(B)

Solution: Let ε>0, a is in A and b is in B,

supA<=a+(ε/2), supB<=b+(ε/2)

sup(A + B) ≥ a + b ≥ sup A − ε /2 + sup B − ε /2 = sup A + sup B − ε.

My Question:

1. Why the answer is not sup(A + B) ≥ a+(ε/2)+b+(ε/2)=a+b+ε?

2. I don't understand why supA<=a+(ε/2), supB<=b+(ε/2)

is a+(ε/2) outside the upper bound or inside the lower bound? why upperbound supA will less and equal to a+(ε/2). (please better draw the horizontal line and explain it)

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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