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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 $\displaystyle \inf_{n \geq 1}(a_n b_n) > (\inf{n \geq n} a_n)b$

in Real Analysis by Expert (2.4k points) | 21 views

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