given the function f: X --> R+ that is bounded above, proove that sup(f^2)=sup^2(f)
I did the proof when given two functions f and g, we have that sup(f*g) is less then or equal to sup(f) * sup(g) using the same hypotheses.
How may I proove the equality when I have one function only?
f: set of all values of f
f2: set of all values of f*f
Briefly. Let x ϵ f, and note there is a real number a between any two real numbers.
1) (lubf)2<a<lubf2 →
x2<a all x, and a<x2 for some x. contradiction.
2) lubf2<a<(lubf)2 →
x2<a all x and a<x2 for some x. contradiction.
Details of 1):
let x ϵ f. Then x ≤ lubf and x2 ≤ (lubf)2 all x → x2 < a all x.
let x ϵ f. Then a < x2 for some x, otherwise a ≥ x2 for all x and a is an upper bound of f2 less than lubf2.