Polynomial versus exponential growth ($n^k=o(a^n)(a>1)$)

Every fixed power is eventually smaller than an exponential function with base $a>1$: for a fixed nonnegative integer $k$, $n^k/a^n\to 0$. To prove this, the ratio of successive terms is $(1+1/n{)}^k/a\to 1/a<1$, so after finitely many terms the ratio is bounded by a fixed $q<1$. A geometric series bound then proves the limit. In particular $n^k=O(a^n)$ in big O notation.