Let $X_1, ..., X_n \stackrel{iid}{\sim} N(0,1)$, define $\hat F_n(x) \equiv \frac{1}{n} \sum_{i=1}^n \mathbf{1}(X_i \leq x)$ and suppose $a_n$ is a sequence of (deterministic) numbers such that
$$a_ne^{\frac{a_n^2}{2}} = n \quad \quad (1)$$
I want to
(i) show that $\hat F_n(a_n) \xrightarrow{P} 1$ and
(ii) determine the limiting distribution of $n(1- \hat F_n(a_n))$.
I don't really have an idea of how to do this properly, but I've tried the following:
For (i): $a_n \rightarrow \infty$, for if it is bounded or goes to $-\infty$ we can't have the LHS in (1) equal to $n$. Now let $\epsilon > 0$ and choose $M \in \mathbb{R}$ such that $1-\Phi(M) < \epsilon$ where $\Phi(x) \equiv \int_{-\infty}^x \frac{e^{\frac{-x^2}{2}}}{\sqrt{2 \pi}}dx$. There must exist an $N$ such that $n \ge N \implies a_n > M$, in which case $1 \geq \hat F_n(a_n) \geq \hat F_n(M) \xrightarrow{a.s.} \Phi(M) > 1- \epsilon$
where the almost sure convergence is by Glivenko-Cantelli and then $\hat F_n(a_n) \xrightarrow{a.s.} 1$ which is even stronger than what I need. Does this work?
I have no idea where to start for (ii). Please help if you can!
What you did for the first part is correct.
For the second one, we have to find the limiting distribution of $Y_n:=\sum_{i=1}^n\mathbf 1\{X_i\gt a_n\}$. Using independence, the characteristic function of $Y_n$ is $$ \varphi_n(t)=\left(1+\left(e^{it}-1\right)\mathbb P\{X_1>a_n\}\right)^n.$$ Using the fact that $a_n\to+\infty$ and the lower and upper tail for Gaussian distribution, we find that $n\mathbb P\{X_1>a_n\}\to 1/\sqrt{2\pi}$. We can conclude using the fact that $(1+x_n/n)^n\to e^x$ if $x_n\to x\in (0,+\infty)$.