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?
Definitions:
f: set of all values of f
f^{2}: 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<lubf^{2} →
x^{2}<a all x, and a<x^{2} for some x. contradiction.
2) lubf^{2}<a<(lubf)^{2} →
x^{2}<a all x and a<x^{2} for some x. contradiction.
Therefore: (lubf)^{2}=lubf^{2}
Details of 1):
(lubf)^{2}<a:
let x ϵ f. Then x ≤ lubf and x^{2} ≤ (lubf)^{2} all x → x^{2} < a all x.
a<lubf^{2}:
let x ϵ f. Then a < x^{2} for some x, otherwise a ≥ x^{2} for all x and a is an upper bound of f^{2} less than lubf^{2}.