If $k$ is any real and $a>1$, prove that there exists a $c>0$ such that for any integer $n\ge 1,$ $$ n^k \le c\cdot a^n $$ To forestall any complaints about the imperative nature of this question, let me give a bit of context. I'm not simply interested in finding a proof; I can give the proof since I can find a suitable value of $c$, given $k$ and $a$. The real question I'm asking is,
How can I provide a proof that students in a 100-level Discrete Mathematics course will understand?
In other words, I expect my students to know pre-calc algebra, but nothing more: no limits, no calculus, so no max/min results, (little or) no understanding of power series, no useful identities like Bernouilli's, and so on. Basically, I can only rely on what they bring from their U.S. high school algebra course.
In the past, I've weaseled out by saying that the proof of the very handy result that $n^k=O(a^n)$ is beyond the scope of the course, but when I sat down to update my lecture notes this summer I wondered whether there was some proof that I'd simply missed. Can anyone come up with one?