Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $a_i$ and $b_i$, $i=1,\ldots,n$, be two finite sequences of numbers in $(0,1)$. We know that $a_i < b_i$ for all $i$. Is it then true that $a_1\cdots a_n < b_1\cdots b_n$?

If all the $a_i$ were the same, and all the $b_i$ were the same, this would be rather obvious since the functions $x^n$ for positive $n$ are known to be increasing on $(0,1)$. So in this special case, it works. But I'm not sure how to generalize this result.

share|cite|improve this question
up vote 5 down vote accepted

If $x>y>0$ and $u>v>0$, then we always have $xu>xv$ and $xv>yv$, so that $xu>yv$.

share|cite|improve this answer

$a_1\cdots a_n<a_1\cdots a_{n-1}b_n<\cdots < a_1\cdots a_{k-1}b_k\cdots b_n<\cdots <a_1b_2\cdots b_n<b_1\cdots b_n$.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.