Problem 2.15 from "Mathematische statistiek" from A. van der Vaart, available here.
For each $n$ let $U_n$ be uniformly distributed over the unit sphere $\mathbb S^{n-1}$ in $\mathbb{R}^n$. Show that the vectors $\sqrt{n}(U_{n,1},U_{n,2})$ converge in distribution to a pair of independent standard normal variables as $n\rightarrow \infty$ [Use the previous problem].
In the link one can find the previous problem 2.14, but it is also here:
Let $Z_1,...,Z_n$ be independent standard normal variables. Show that the vector $U=(Z_1,...,Z_n)/N$, where $N^2 = \sum\limits_{i=1}^n Z_i^2$, is uniformly distributed over the unit sphere $\mathbb S^{n-1}$ in $\mathbb{R}^n$ in the sense that $U$ and $OU$ are identically distributed for every orthogonal transformation $O$ of $\mathbb{R}^n$.
I don't see the link between problem 2.15 and 2.14. In problem 2.15 we have to show something about independent standard normal variables given some variables uniformly distributed in the unit sphere, and problem 2.14 is the opposite, there we have to show that a vector is uniformly distributed.
Further convergence of $\sqrt{n}$ times a vector makes me think of the multidimensional central limit theorem, but I can see no average.
Can anyone help?