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.