Can we establish the following inequalities

Let $A$ and $B$ both be sets of real numbers. Let $A \times B$ be $\{xy \mid x \in A, y \in B\}$. Can we establish the following:

$\sup A \sup B \leq \sup(A\times B)$

and

$\inf A \inf B \geq \inf(A\times B)$

I've been trying hard to come up with counterexamples but I haven't been able to come up with any. Are those inequalities true?

Choose $(a_{n})\subseteq A$, $(b_{n})\subseteq B$ such that $a_{n}\rightarrow\sup A$ and $b_{n}\rightarrow\sup B$, then $\sup(AB)\geq a_{n}b_{n}$, taking limit as $n\rightarrow\infty$, then $\sup(AB)\geq(\sup A)(\sup B)$.

If we define $0\cdot\infty=0$, it still goes through: For $\sup A=\infty$ and $\sup B=0$, then choosing $a\in A$, $a>0$, $(b_{n})\subseteq B$, $b_{n}\rightarrow 0$, then $\sup(AB)\geq ab_{n}$, taking limit as $n\rightarrow\infty$, we have $\sup(AB)\geq 0=(\sup A)(\sup B)$.