For natural numbers, I would like to prove that if $m < n$, then $k^m < k^n$ for $k$ not $0$ or $1$. Induction seems viable, but I don't know which variable to induct on. Any suggestions?
If m < n, then k^m < k^n?
2
$\begingroup$
induction
-
1It would be very helpful if you could please elaborate on your earlier comment that you are "developing naturals set theoretically", so that we know what methods are available to you. The answer could vary widely depending on the foundational level that you are working at. If the question is from a book then a simple reference would suffice to convey that contextual information. – 2011-03-02
1 Answers
2
Hint: Consider $f(x)=a^{x}$ and prove that it's monotonically increasing.