Using Central Limit Theorem to Prove Claims About Distributions? (Desperate) I am kind of confused for the following questions:

I know for the second question that I have to use the delta method but I can't even get past the first question here. Like, my thinking initially was that the CLT takes the form of $\sqrt n(\bar{x} - \mu)$ -> ... and so, I know that here, I am minimizing the difference between the sd observed and the variance expected for this distribution (the normal one), but I know that the distribution of the variance is chi squared, so the $\bar{x}$ if I change it over to that dist is $s^2$ in this question and same with the $\sigma^2$. However, then the expected value there is $n-1$ and since n tends infinity.. this obviously doesn't make sense. Can someone give me a good understanding of questions like these? My final is coming up and need help! Thank you.
 A: Recall that 
$$s^2 = \frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2 = \frac{1}{n-1}\sum_{i=1}^nX_i^2 - \frac{n}{n-1}\bar X^2$$
Now look at the first term and think whether you can apply CLT to it. Then think what is the limit of the second term in terms of convergence of probability. Recall Slutsky lemma and think how can it help you. 
Remember that the CLT in its simplest form is stated as the following. For iid random variables $\xi_i$ with $E\xi_i=\mu$ and $Var(\xi_i)=\sigma^2$ the following holds:
$$\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^n\xi_i - \mu\right) \xrightarrow{D} N(0,\sigma^2).$$
Also it is useful to remember that LLN applies for the same random variables:
$$\frac{1}{n}\sum_{i=1}^n\xi_i \xrightarrow{P} \mu.$$
Another useful fact is the continuous mapping theorem which states that if
$$\eta_n\to \eta,$$
then 
$$g(\eta_n)\to g(\eta),$$
where $\eta_n$ is a sequence of random variables and $g$ is a continuous function. The convergence in this case can be both in distribution and in probability. Do not confuse it with delta method though!
A: I give you some suggestions to solve (a).
Remember $\sqrt{n}(\hat{\theta}-\theta_0)\overset{D}{\rightarrow} N(0,\frac{1}{I(\theta)})$, you will use CLT here.
Here $\hat{\theta}=s^2$ and $\theta_0=\sigma^2$
Next calculate Fisher informtion from $N(\mu,\sigma^2)$ treat $\sigma^2$ as $\theta_0$, then you will get $\psi$
