Question:
$\alpha \in \mathbb{R}$, if $n^{\alpha}$ is an integer for $\forall n \in \mathbb{N}$, then show that $\alpha$ is a nonnegative integer.
I totally don't know how the procedure should go. Besides, I got this problem in the calculus textbook, but I cannot understand why the problem is included in the textbook.
Can you give me some simple key points to this kind of problem? Thanks for your advice.
[EDIT] I modified some expressions in order not to have various interpretations.