Rational Solver
0 votes
105 views

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 | 105 views

Please log in or register to answer this question.

Welcome to Rational Solver, where you can ask questions and receive answers from other members of the community.
65 questions
23 answers
2 comments
1,801 users