After proving $m^n \geq n^m$ for several values of $m$, it can be inferred that for every $m$ there's a $k$ such that if $n \geq k$, $m^n \geq n^m$.
In other words, this can be generalized as:
For all $m$, there's a $k$ such that for all $n$ if $n\geq k$, $m^n \geq n^m$.
The problem is, I don't know how to start.
I know $m>1$ for it to work, and this sounds inductive... however; how can I know what value of $k$ should work for a general case? Should I try induction by cases...
Assume $m$ is a natural, prove something and then conclude that $m$ has to be $m>1$? Or should I assume $m>1$ and then proceed to prove that there's a $k$ for every m such that $n \geq k$ implies $m^n \geq n^m$.
Thanks for any advice or hints.
Two hints:
$m^n \ge n^m \iff m^{1/m} \ge n^{1/n}$
the function $f(x)=x^{1/x}$ is decreasing for $x>e$