Let $A,B\subseteq \Bbb R $such that A and B are bounded, show that AB is bounded $AB=\lbrace ab \mid a\in A , b\in B \rbrace$.
Let $M$ is an upper bound of $A$ and $N$ is an upper bound of $B$.
Then can we say that $MN$ is upper bound of $AB$?
So how can we prove that $AB$ is bounded?

$A, B$ are bounded, let $S,T$ be the bounds for $A,B$ resp., i.e.
$|a|\lt S$,  $a \in A$, $|b| \lt T$, $b \in B$.
$AB:=${$ab| a \in A, b \in B$}
For $ab \in AB$:
$|ab|=|a||b| \lt ST$, i .e. $AB$  is bounded.
