How to show that $\lim_{n\to\infty} c^n\sqrt n = 0$, where $c\in(0,1)$? 
How to show that $\lim_{n\to\infty} c^n\sqrt n = 0$, where $c\in(0,1)$? 

We often claim that exponent is "stronger" than root (and we even proved it last semester) but I can't remember how.

$c^n\sqrt n=e^{n\ln c}\sqrt n=e^{n\ln c+\ln\sqrt n}$ but 
$n\ln c+\ln\sqrt n=n(\ln c+\dfrac{\ln \sqrt n}{n})\to -\infty$ ((because $\dfrac{\ln \sqrt n}{n}\to 0$ and $\ln c<0$)). Hence $c^n\sqrt n\to 0$. 
