Let $L_k$ uniform (discrete) i.i.d. variables in $\{0,1\}$. How to prove $$X:=\sum_{k=1}^\infty \frac{L_k}{2^k}$$ is uniformly distributed in $[0,1]$?
Of course I have to show that the cdf is the same, which means I have to prove for all $n \in \mathbb{N}$, $$P\left(X \leq \frac{j}{2^n}\right) = \frac{j}{2^n}.$$
I have no idea how to continue after
$$P\left(X \leq \frac{j}{2^n} \right) = P\left(\sum_{k=1}^\infty \frac{L_k}{2^k}\leq \frac{j}{2^n} \right)$$
Can sb. give me a hint?