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?

No $MN$ won’t be the upper bound of $AB$. Take the example of $A=[-1,0]$ and $B=\{1\}$.
However if $m$ is such that $\vert a \vert \le m$ for all $a \in A$ and $n$ is such that $\vert b \vert \le n$ for all $b \in B$, $mn$ is such that $\vert x \vert\le mn$ for all $x \in AB$. That proves that $AB$ is indeed bounded.
