Monotone Sequences and Their Transformations: Proving or Disproving Monotonicity

[cnwilson2010]

Homework Statement

Let a_n be monotone sequences. Prove or give a counterexample:

The sequence c_n given by c_n=k*a_n is monotone for any Real number k.

The sequence (c_n) given by c_n=(a_n/b_n) is monotone.

Homework Equations

The Attempt at a Solution

On the first one, I don't think the change of sign on k can change the "monotoneness" of the sequence other than by changing decreasing to increasing and vice versa.

I have played around using different sequences to see if this is true and it is looking like it is, but I just feel that it could be false.

Any ideas?

 

[lanedance]

for the first calculate the difference between terms and show it is always either pos or neg

2nd find a simple counter example, consider an alternating series

 

[cnwilson2010]

I neglected to put the condition that b_n is also monotone.

So I was thinking of a_n with a different sign than b_n but this doesn't seem to change much either.