Let be $X_k$ independent random variables, where $\mathbb P\left(X_k=\pm2^{-k}\right)=1/2$. Prove that $$S_n=\sum_{k=1}^{n}{X_k}\rightarrow UNI(-1,1).$$
It is easy to see with the help of characteristic functions, and using the following identity: $$\frac{\sin{t}}{t}=\prod_{k=1}^{\infty}\cos{\frac{t}{2^k}}.$$
My question is, how can I prove it without characteristic functions?
Note: One can say, that this is a probabilistic proof of the above mentioned trigonometric identity.
By induction, $S_n$ has a discrete uniform distribution on the set $(j/2^n)$ where $j$ ranges over $J_n$, the odd integers with $-2^n<j<2^n.$ Then for a continuous function $f$ on $[-1,1]$, we have $$\mathbb{E}(f(S_n))=\sum_{j\in J_n}f\left({j\over 2^n}\right){1\over2^n}\to{1\over 2}\int_{-1}^1 f(x)\,dx,$$ since the Riemann sums converge to the integral.
This shows that $S_n$ converges in distribution to a uniform$(-1,1)$ random variable.