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Let {an}, {bn} be given bounded sequence of positive real numbers. Then (Here ana means an increases to a as n goes to , similarly, bnb means bn increases to b as n goes to

  1.  If ana, then sup
  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

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