1. ## abstract algebra

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?

2. ## Re: abstract algebra

Originally Posted by Yeison
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?
From that, you know that $\sup(f^2)\le\sup^2(f)$. Right?

What if $\sup(f^2) < \sup^2(f)~?$

3. ## Re: abstract algebra

Definitions:
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.

Therefore: (lubf)2=lubf2

Details of 1):
(lubf)2<a:
let x ϵ f. Then x ≤ lubf and x2 ≤ (lubf)2 all x → x2 < a all x.

a<lubf2:
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.