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.

The sequence is positive, so
$$\sqrt[n]{c^n\sqrt n}=c\sqrt{\sqrt[n]n}\xrightarrow[n\to\infty]{}c<1$$ 
and by the $\;n$- th root test the infinite series
$$\sum_{n=1}^\infty c^n\sqrt n\;\;\;\text{converges}\;\;\implies c^n\sqrt n\xrightarrow[n\to\infty]{}0$$
