Question

If a ≥ 0 and b ≥ 0, prove that a^n ≥ b^n, if and only if a ≥ b. (n is a positive integer)

Answer 1

First, start with a ≥ b.

In which case ab ≥ b^2 but a^2 ≥ ab, by hypothesis, so a^2 ≥ b^2

Now you should be able to see how to prove from our hypothesis that a^n ≥ b^n for all positive integers n.

That's one direction of the "if and only if"

We now need to prove that a^n ≥ b^n => a ≥ b. It will be easier to prove the contrapositive: a < b => a^n < b^n.

The argument for this last deduction mimics the one we just used.
All clear?

Answer 2

Makes a lot of sense. Thanks. I hadn't thought of the contrapositive to prove the second part. Thanks again.