Regarding $x < y \Rightarrow x^n < y^n$ proof rigor.

Question

I came across the implication $$x < y \Rightarrow x^n < y^n$$ $$x,y>0, n\in Z^+$$ in a textbook and came up with the following proof.

Proof Since $x<y$ the following chain of inequalities holds. $x^n<x^{n-1}y < x^{n-2}y^2 <...<y^n$

My question now relates to the “solidity” of this proof. I get the feeling that it is a bit vague in its reasoning, yet I cannot see that it isn’t correct... I’m quite new to proving things and can’t always see the difference between intuitive reasoning and rigorous reasoning.

Is my reasoning solid? If not, what is it lacking?

Answer 1

What's lacking is explicit induction. It's an "et cetera" argument. It's not wrong, but it's not $100\%$ rigorous. It's close enough that making it rigorous is easy though, if you know how to do induction.

Answer 2

Ask yourself the following question: Where in the proof do I (tacitly) use the hypothesis that $x$ and $y$ are positive? Then make that use explicit.

It might also help to rewrite the proof as a proof by induction, inducting on $n$. In other words, show that if $0\lt x\lt y$ and $x^n\lt y^n$, then $x^{n+1}\lt y^{n+1}$ (again, making explicit where the positivity hypothesis is used).

Remark: In general, it's a good idea, when either writing or reading a proof, to ask, what are the hypotheses and where in the proof are they used? It sometimes happens that a proof does not use a hypothesis, even tacitly, which sometimes means that the hypothesis was unnecessary -- and sometimes means the proof is wrong!