I am working on a larger proof, but for my chosen method, I need to know if the following is true:

If $b_n>0$ for all $n\in\mathbb{N}$, and $b_n\leq b_1$ for all $n\in\mathbb{N}$, then is it true that $|a_1b_1+\dots a_nb_n|\leq|a_1b_1+\dots+a_nb_1|$?

I've tried to use an induction proof and got stuck. I'm starting to think it is not true, although I haven't found a counterexample yet.

Thanks for your help.

  • 2
    $\begingroup$ If the $a_n$ involved are all nonnegative, it's true. Otherwise, not. For example, $n=2$, $a_1=1$, $a_2=-1$, $b_1=2$, $b_2=1$. $\endgroup$ – Harald Hanche-Olsen Sep 30 '14 at 14:24
  • $\begingroup$ Thanks for the comment. I edited the post to reflect the fact that the $b_n$ terms are in fact all positive, but we know nothing about the $a_n$ terms. So we can't guarantee it then? $\endgroup$ – nonremovable Sep 30 '14 at 14:26
  • $\begingroup$ I updated my comment before seeing your answer. Sorry if it messes up the discussion thread a bit. $\endgroup$ – Harald Hanche-Olsen Sep 30 '14 at 14:26

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