im struggling to prove the following problem.
Let $A \subseteq \Bbb R$ be a non empty set, (multipication is for example $A\cdot B= \{a \cdot b\ |\ a \in A,\ b \in B\}$)
Show: If $A$ is bounded, then $A\cdot A$ is bounded too!
I know that from the completenes axiom the set A has an infimum and a supremum. what should I do next . I'm new to this material and i dont understand it to it's fullest.
pls help, thanks
If a $A \subseteq \mathbb{R}$ is bounded then there exists a positive number $r \in \mathbb{R}_{+}$ such that $A$ is contained in the ball $B_r(0)$ with radius $r$ centered in $0$. Clearly, $$ A\cdot A \subseteq B_r(0) \cdot B_r(0) \subseteq B_{r^2}(0). $$ For the last inclusion take $x,y \in B_r(0)$. Then we have $|x|, |y| < r$, which implies $$|x\cdot y| = |x|\cdot |y| < r\cdot r.$$ So $A\cdot A$ is bounded by $r^2$.
This proof can be easily reused for bounded sets in $\mathbb{C}$.