Proof that $\lim_{n\to\infty} n\left(\frac{1}{2}\right)^n = 0$ Please show how to prove that $$\lim_{n\to\infty} n\left(\frac{1}{2}\right)^n = 0$$

By the Binomial Theorem, if $n\ge 2$ then
$$(1+1)^n \ge 1+n+\frac{n(n-1)}{2}\gt \frac{n(n-1)}{2}.$$
It follows that if $n\ge 2$ then $\dfrac{n}{2^n}\lt \dfrac{2}{n-1}$. 
