Having a hard time understanding this

[STEMucator]

Prove that if $m > 1$ such that there exists a $c > 1$ that satisfies
$$cm < m^c$$
then for any $k > c$
$$km < m^k$$
holds. Prove this without using logarithms or exponents or calculus. Basically using the properties of real numbers to prove this.

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?

 

[Arkuski]

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.

 

[LeonhardEuler]

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.

 

[STEMucator]

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.

Yes this makes sense now, thank you. If that were the case though, would I have been wrong to say that?

I thought he meant for every m>1, there exists a c>1 which satisfies .. blah.