Show that a set given is bounded

Question

Given a bounded set $A$. Show that $aA = \{ax | x \in A\}$ and $a \in \mathbb{R}$ is also bounded.

The question is looked simple at least. Since $A$ is bounded, it must be have supremum and infimum. But I have a howler when my friend said that $a$ could be a negative number and we can't rely on that $aA$ is bounded above by $a \sup (A)$. Beside that, $\sup(A)$ also can be negative. I have idea to take it account as some cases, for positive and negative, proving that every cases leads to a conclusion that $aA$ is bounded, but it seems complicated.

Do you have any good idea or the way to solve this problem?

Answer 1

Guide:

Since $A\subseteq\mathbb R$ is bounded some $r\in(0,\infty)$ will exist with: $$A\subseteq[-r,r]$$ and consequently: $$aA\subseteq a[-r,r]$$

Now prove that: $$a[-r,r]=[-|a|r,|a|r]$$

Answer 2

As $A$ is bounded, we see there exists a $r>0$ such that \begin{align} A \subset B_r(0) \end{align} where $B_r(0) = \{x : |x| < r\}$. Now we fix $a \in \mathbb{R}$, we see as $a \in \mathbb{R}$, $|a| < \infty$. In particular, as $aA = \{ax:x \in A\} \Rightarrow aA \subset B_{ra}(0)$. Hence, we see $aA$ is bounded.