In outline: one approach is to think of generating $U_n$ by generating $n$ iid standard Normals $Z_{n,1},\ldots,Z_{n,n}$ and defining $$U_{n,i}=\frac{Z_{n,i}}{\sqrt{\sum_j Z_{n,j}^2}}$$
As $n\to\infty$, the denominator converges to its expected value (eg, by Chebyshev's inequality) and can be treated as a constant. The expected value is a multiple of $\sqrt{n}$, so rescaling any finite set of $U_{n,i}$ by $\sqrt{n}$ will asymptotically give independent Gaussians that are just multiples of the corresponding $Z_{n,i}$.
Update: the result is fairly straightforward but the implications are non-intuitive. $U_{n,1}= O_p(n^{-1/2})$, for $U_n$ uniformly distributed on $S^n$, so nearly all of the area of $S^n$ is within $O(n^{-1/2})$ of the equator for large $n$(!!).