I am reading "Asymptotic Statistics" by A.W van der Vaart and I am stuck with an exercise of chapter 2.
Here is the question : for each $n \in \mathbb{N}$, let $U_n$ be uniformly distributed on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$. Show that the random vectors $\sqrt{n}(U_{n,1},U_{n,2})$ converge in distribution to a pair of independent standard normal variables.

Maybe the solution is extremely stupid but I don't know where to start. Could you provide me some hint ?

Also, I am sorry if the solution is already available on the internet, I couldn't find it.