Real analysis question about limits? If $0  < x < 1$, show that $x^n → 0$ as $n → ∞$ .
I'm thinking it has something to do with rational numbers (declaring $x = p/q$) and then using exponent laws to show that $x^n = p^n/q^n$, and since $q > p$ ... but I'm not sure where to go with this.  Any help is appreciated.  Thanks

Hints:
$$0<x<1\implies x>x^2>x^3>\ldots >x^n\;,\;\;\forall\,n\in\Bbb N$$
$$a_n>0\;\;\forall\,n\in\Bbb N\implies \lim_{n\to\infty}a_n=\infty\iff \lim_{n\to\infty}\frac1{a_n}=0$$
