• Jun 30th 2011, 08:09 AM
CountingPenguins
The question is:

Let ( $a_n$) and ( $b_n$) be monotone sequences. Prove or give a counterexample.

a. The sequence ( $c_n$) given by ( $c_n$)= k* $a_n$ is monotone for any real number k.

I don't think the change of sign on a constant will do more than turn a monotone increasing sequence to a monotone decreasing sequence so this statement is probably true.

b. The sequence ( $c_n$) given by ( $c_n$)= $a_n$/[tex]b_n[tex] is monotone.

I'm also pretty sure that having opposite signs for the sequences being divided isn't going to change much either. So I suspect I need to show the different cases on this proof is that the general idea?
• Jun 30th 2011, 09:44 AM
FernandoRevilla
The statement a. is true. However, b. is false (even supposing $a_n/b_n$ well defined i.e. $b_n\neq 0$ for all $n$ ) . Choose for example $a_1=1,a_2=2$ , $a_n=3$ for all $n\geq 3$ and $b_1=1$ , $b_n=3$ for all $n\geq 2$ .