Hello, first time poster here. I've recently entered a Discrete course at my university and the professor does not exactly line up with my style of learning. I like to believe that I learn by example, and he's more of a "here's the rules, and now I let you go".

So I was wondering if you could assist me with this one question, and from there I'd be good. Here it is:

Prove that for any real number

, if

, then for all positive integers

and

, if

, then

>

. Use mathematical induction.

First off, I understand the general concept that since r is a positive real number less then 1, that the larger it's exponent, the smaller the calculated number will be. I also vaguely understand the idea of a predicate, which I believe is required for the concept of mathematical induction, being that first I must prove the base case, then a general case, then the step after any general case. What confuses me however, is just the general application of all this, specifically since I think this question involves a predicate with multiple variables, something I have not dealt with yet.

If anyone were to be able to walk me through this question, I would be very grateful! Thank you.