Prove that $\lim_{n \to \infty} k^nn^p= 0$ where $|k| < 1$ and $p>0$ Hello I am working through some problems in a book and came across this question.
Prove that $\lim_{n \to \infty} k^nn^p= 0$ where $|k| < 1$ and $p>0$ 
I can see why it should be the case (since exponentials grow faster than polynomials) but I don't really know where to start to try and prove it rigourously. I have identified that it becomes a $"0 \times \infty"$ situation and have tried using L'Hôpital's rule on the expression $k^n/(1/n^p)$ which didn't seem to help. How should go about tackling this problem?

It suffices to show 
\begin{align*}
(|k|^{1/p})^{n}n\rightarrow 0.
\end{align*}
Let $a=|k|^{1/p}<1$, we are to show that $a^{n}n\rightarrow 0$.
Let $a=1/(1+r)$ for $r>0$, then $a^{n}\leq\dfrac{1}{1+nr+n(n-1)r^{2}/2}$, it is now easy to show that
\begin{align*}
\dfrac{n}{1+nr+n(n-1)r^{2}/2}\rightarrow 0.
\end{align*}
