Consider $(X_{1},\ldots,X_{n})$ a sequence of random variable i.i.d such as $P(X_j=1)=P(X_j=-1)=\frac 12$ for all $j \geq 1$. Consider now the sequence $Y_{n} = \sum_{j=1}^{n} 2^{-j} X_{j}$ for all $n \geq 1$. Proof that $Y_{n}$ converges in distribution to $\operatorname{Unif}(-1,1)$.
It's quite easy to proof that $P(Y_n \leq -1) = P(Y_n\geq 1) = 0$. But how can i get $P\left(\sum_{j=1}^{n} 2^{-j} X_j \leq z\right)$ for $z \in [-1,1]$?
Thanks in advance for any help.