Hello, I need help with the develpment of this exercise that I'm trying to solve.

Let \widehat{\sigma} be the U.M.V.Uestimate of \sigma when \mu is unknown
show that:
a) \sqrt{n}{(\widehat{\sigma}-\sigma})\rightarrow\,{N(0,\sigma^2/2)}

b) \displaystyle nVar_\theta\rightarrow{\sigma^2/2},\sqrt{n}b(\sigma,\widehat{\sigma})\rightarrow{0}

hints:
a)\sqrt{n}{(\widehat{\sigma^2}-\sigma^2})/\sigma^2\approx {1/\sqrt{n} \displaystyle\sum^n_i=1(Z_i^2-1)} where the \displaystyle Z_i^2 are independent \chi^2_1

b) Use the stirling's approximation, \lim_{x\rightarrow \infty} \Gamma(x + 1)/(\sqrt{2\pi}e^{-x}x^{x + \frac{1}{2}})=1


note: the estimator \widehat{\sigma} is defined by:
\widehat{\sigma}=\sqrt{\frac{1}{n}\displaystyle\sum^n_{i=1}(X_i-\bar{X})} where X_i is a sample form a \mathbb N(\mu,\sigma^2)