I'd like to prove that $a^n + b^n = c^n$ where $n>2$ and $a>0$, $b>0$, and $c>0$ has no solution.
I first divided each by $a$, then I get $c>a$.
Then, I divided each by $b$, then I get $c>b$.
Finally, I divide each by $c$, then I get $(a/c)^n + (b/c)^n = 1$.
Because if $n$ infinitely increases, $(a/c)^n + (b/c)^n$ is $0$. Thus, there are no values that satisfy the equation.
I think my proof is somewhat awkward. Can anybody help solve this?
Your attempt of proof has a problem in quantification. The problem is in this passage:
"Because if $n$ infinitely increases, $(a/c)^n+(b/c)^n$ is $0$''. You can only do this if $\forall n>2:(a/c)^n+(b/c)^n=1$, while you only have the hypothesis $\exists n>2:(a/c)^n+(b/c)^n=1$.
In other words, what are you trying to prove is:
If $\exists a,b,c>0:\exists n>2:a^n+b^n=c^n$, then $\bot$,
where $\bot$ stand for contradiction. But you only proved that:
If $\exists a,b,c>0:\forall n>2:a^n+b^n=c^n$, then $\bot$.
