Let $\{a_n\}$, $\{b_n\}$ be given bounded sequence of positive real numbers. Then (Here $a_n \uparrow a$ means $a_n$ increases to $a$ as $n$ goes to $\infty$, similarly, $b_n \uparrow b$ means $b_n$ increases to $b$ as $n$ goes to $\infty$)
1.  If $a_n \uparrow a$, then $\displaystyle \sup_{n \geq 1}(a_n b_n) = a(\sup{n \geq n} b_n)$
2. If $a_n \uparrow a$, then $\displaystyle \sup_{n \geq 1}(a_n b_n) < a(\sup{n \geq n} b_n)$
3. If $b_n \uparrow b$, then $\displaystyle \inf_{n \geq 1}(a_n b_n) = (\inf{n \geq n} a_n)b$
4. If $b_n \uparrow b$, then $\displaystyle \inf_{n \geq 1}(a_n b_n) > (\inf{n \geq n} a_n)b$