Let $f,g: A \to [0,\infty)$ be two functions defined in a non-empty set $A \subset \mathbb R$.
- Prove that if $f$ and $g$ are bounded above, then $f \cdot g$ is bounded above and $\sup(f \cdot g) \leq \sup f \cdot \sup g$.
- Give an example where $\sup(f \cdot g) \lt \sup f \cdot \sup g$ is valid.
My attempt
(1) Let $C = \{ g(x) \cdot f(x) : x \in A\} , B = \{g(y): y \in A\}$ and $A = \{ f(x): x \in A\}$.Taking $x=y$ in the definitions we know that $C \subset A \cdot B$ is valid, then
$$ \begin{split} \sup (A \cdot B) &\geq \sup (C) \\ sup(A) \cdot \sup(B) &\geq \sup(C) \\ \sup(f) \cdot \sup(g) &\geq \sup(f \cdot g) \end{split} $$
I can't think of anything for (2). I appreciate in advance for any help.
For $2)$ let $f(x)$ map $[0,1]\to 1$ and $(1,2]\to 0$ and let $g(x)$ be a function which maps $[0,1]\to 0$ and $(1,2]\to 1$