Least Upper Bound: Understanding sup(f+g)<=sup(f)+sup(g)

[omri3012]

Hallo,

i read in Spivak that for every upper bounded non empty sets A and B,

sup(A+B)=sup(A)+sup(B). but later he wrote other prove which claim that

for every function f and g in a close interval exist sup(f+g)<=sup(f)+sup(g)

and not necessarily sup(f+g)<=sup(f)+sup(g). how does it make sense?

Thanks

Omri

 

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A+B is the set of all possible sums between elements of A and elements of B.
Unfortunately, the set pertaining to the terms (f+g) is defined pointwise; it only contains elements of the form (f+g)(x), so suppose f and g are defined on [a, b] and sup(f) occurs at a and nowhere else in the interval and sup(g) occurs at b and nowhere else in the interval. Then sup(f+g) never equals sup(f) + sup(g).
If, on the other hand, we were talking about all possible sums of elements in the range of f with elements in the range of g, then of course sup(f) + sup(g) would be one of those elements.