1. ## Proof by induction

I have this question and I think i've done it. The trouble is, thinking i've done it isn't enough and I need to know if I actually have!

Question: Use induction to prove that if both x and y are positive then $\displaystyle x<y \Rightarrow x^n<y^n$.

I did this:

I rearranged it like so $\displaystyle \frac {x}{y} <1 \Rightarrow \frac {x^n}{y^n} <1$

For n=1:

$\displaystyle \frac {x^1}{y^1}<1 \Rightarrow \frac {x}{y} <1$

Therefore it is true for n=1.

For n=k:

$\displaystyle \frac {x^k}{y^k} <1$

For n=k+1:

$\displaystyle \frac {x^{k+1}}{y^{k+1}}<1$

$\displaystyle \frac {x^k}{y^k} \frac {x}{y}<1$

$\displaystyle \frac {x^k}{y^k}< \frac {y}{x}$

We know that$\displaystyle \frac{y}{x}>1$ and $\displaystyle \frac{x^k}{y^k}<1$ so this inequality makes sense.

At this point I think it has been proven true for n=k+1.

So is that the end of the proof? I still have to write the little thing on the end explaining it was proof by induction. I just want to know if this is the right method.

2. It looks ok in general. The only nit-pick I have is the order of demonstrating n ->n+1. You kind of assumed n+1 was true then justified it after the fact. I would just reverse the order so that it flows more naturally going from n, doing some algebra, and implying n+1.

But yes, once you show n->n+1 you are basically done. You only have to make some sentence wrapping up the proof.

3. Induction seems like a bit of overkill for this problem, as you're raising a number that is less than one to a positive power. This is almost trivial.

Bobak

4. Originally Posted by bobak
Induction seems like a bit of overkill for this problem, as you're raising a number that is less than one to a positive power. This is almost trivial.

Bobak
True. But if a question prescribes a method, then that's the method that has to be used. Especially important to remember this in exams.

5. It looks ok in general. The only nit-pick I have is the order of demonstrating n ->n+1. You kind of assumed n+1 was true then justified it after the fact. I would just reverse the order so that it flows more naturally going from n, doing some algebra, and implying n+1.
Okay great! It's nice to know that i'm doing it correctly.

What do you mean "reverse the order"?

Do you mean I should write something like "assume true for n=k" or do you mean I should write $\displaystyle \frac{x^{k+1}}{y^{k+1}}$ and not put the <1 part in?

Induction seems like a bit of overkill for this problem, as you're raising a number that is less than one to a positive power. This is almost trivial.
As Mr Fantastic said, it does say to use this method. I was also thinking it would just be easier to say that $\displaystyle \frac {x}{y}$ is less than one (but greater than 0). When a fraction is raised to a positive power it would get smaller but instead I have to do it a long way.