This question was posted in the homework forum. To my understanding, this looks like two different questions as the first hypothesis is not true due to a simple counter-example argument. The second one would be true by two induction arguments on k>2 and k<0.

Am I wrong in thinking this? It seems logical since he said "there exists a c >1" so I would figure any REAL c must make the inequality true?

I would use induction on this. The wording is pretty awkward and it seems like two different questions but I think they are looking for an induction type proof.

Zondrina, I had the same confusion as you when I first read the question. It is actually only one question. The first part is not saying
" if m>1 then there exists a c>1 that satisfies
cm<mc"
The whole thing is saying that if the number "m" is such that some "c" exists satisfying the requirement, then every k>c satisfies the second inequality.