analysis - Prove $displaystylelim_{nto infty}{nb^n}=0$

Prove $\displaystyle\lim_{n\to \infty}{nb^n}=0, 0

One last thing I used a certain theorem in my proof which I will give.

Theorem 3.1.10 Let $(x_n)$ be a sequence of real numbers and let $x\in \mathbb{R}$ If $(a_n)$ is a sequence of positive real numbers with $\lim(a_n) = 0$ and if for some constant $C > 0$ and some $m \in\mathbb{N}$ we have $|x_n-x|\leq Ca_n$ for all $n > m$,
then it follows that $lim(x_n ) = x$.

Here is how my proof goes. Though I don't think its right.

Since $00$and$n \in
\mathbb{N}$It follows that$nb^n=\frac{n}{(1+h_n)^n}$. Then by Bernoulli's inequality we get$\frac{n}{(1+h_n)^n}\leq \frac{n}{(1+nh_n)}\leq \frac{n}{nh_n}$By the theorem above we let$C=\frac{n}{h_n}$and$a_n=\frac{1}{n}$. Thus$\displaystyle\lim_{n\to \infty}{nb^n}=0$