A limit about $\prod_{k=0}^\infty\frac1{1-x^k}$

Question

If $$\sum\limits_{n = 0}^\infty {{a_n}{x^n}} = \prod\limits_{k = 0}^\infty {\frac{1}{{1 - {x^k}}}} ,$$ Prove $${a_n} < \exp \left\{ {\sqrt {\frac{{2\pi }}{3}n} } \right\}$$ and $$\mathop {\lim }\limits_{n \to \infty } \frac{{n{a_n}}}{{\exp \left\{ {\pi \sqrt {\frac{{2n}}{3}} } \right\}}} = \frac{{\sqrt 3 }}{{12}}.$$

I have no idea! Please show me how to solve it, thanks!