Let $\mathbf{x}=(x_1,...,x_n)\sim \text{Uniform}(\mathbb{S}^{n-1})$ and $z\sim\mathcal{N}(0,1)$. How to prove for any $A\subseteq \mathbb{R}$ $$\Big|\mathbb{P}\Big\{\sum_i x_i \in A \Big\} - \mathbb{P}\Big\{z \in A \Big\}\Big| \to 0$$ ?
Intuitively, I think the statement is true. Since for $n\to\infty$, $\mathbf{x}$ approximates $\mathbf{y}\sim\frac{1}{\sqrt{n}}\mathcal{N}(\mathbf{0},\mathbf{I}_n)$ as the mess of high-dimensional Gaussian distribution concentrates around the unit sphere scaled by $\sqrt{n}$. And $\sum_i y_i$ is equivalent to $z$.
But can anyone give me a rigorous proof?