please explain the solution and whatit is wrong with my conception. FOLLOW theCOMMENT PLEASE
Question:
Let A and B are nonempty set bounded subset of R, and let A+B bethe 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 inB,
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 lowerbound? why upperbound supA will less and equal to a+(ε/2). (pleasebetter draw the horizontal line and explain it)
