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