Proving r^n > r^m through mathematical induction

[tmay82]

Homework Statement

I need to prove that for any real number r, if 0 < r < 1, then for all positive integers n and m, if n < m, then r^n > r^m.

Homework Equations

No calculus techniques are permitted, only mathematical induction.

The Attempt at a Solution

I know that any fraction between 0 and 1 is going to get smaller if it is multiplied by anything positive, so this is obviously true.

I know that I first need to figure out what predicate to use, but I'm having a problem with all of the variables.

Im not looking for the answer, just a little bit of direction. Where/how do I begin? I know what to prove, I just don't know how to prove it.

Thanks for any help out there

 

[PeterO]

Homework Statement

I need to prove that for any real number r, if 0 < r < 1, then for all positive integers n and m, if n < m, then r^n > r^m.

Homework Equations

No calculus techniques are permitted, only mathematical induction.

The Attempt at a Solution

I know that any fraction between 0 and 1 is going to get smaller if it is multiplied by anything positive, so this is obviously true.

I know that I first need to figure out what predicate to use, but I'm having a problem with all of the variables.

Im not looking for the answer, just a little bit of direction. Where/how do I begin? I know what to prove, I just don't know how to prove it.

Thanks for any help out there

Perhaps you could take the logarithm of both sides of the inequality.

 

[ElijahRockers]

To be honest I don't know much about induction, but it has been something I want to learn. I wasn't going to reply here because I don't know anything about it, but I went to khan academy and found this video! Coincidence? I don't know if it's what you need, but I hope it helps.

http://www.khanacademy.org/video/proof-by-induction?playlist=Algebra

 

[uart]

I know that any fraction between 0 and 1 is going to get smaller if it is multiplied by anything positive, so this is obviously true.

No, you have that the wrong way around. Anything positive is going to get smaller if multiplied by a number between 0 and 1. So that's the basis of your inductive step right there. Now you just need to write it out formally as an inductive proof.