1. ## Question about monotone sequences.

The question is:

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

a. The sequence ($\displaystyle c_n$) given by ($\displaystyle c_n$)= k*$\displaystyle a_n$ 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 ($\displaystyle c_n$) given by ($\displaystyle c_n$)= $\displaystyle a_n$/[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?

2. ## Re: Question about monotone sequences.

The statement a. is true. However, b. is false (even supposing $\displaystyle a_n/b_n$ well defined i.e. $\displaystyle b_n\neq 0$ for all $\displaystyle n$ ) . Choose for example $\displaystyle a_1=1,a_2=2$ , $\displaystyle a_n=3$ for all $\displaystyle n\geq 3$ and $\displaystyle b_1=1$ , $\displaystyle b_n=3$ for all $\displaystyle n\geq 2$ .