Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$ and $F_n(x) = \mathbb{P}(Y_n \leqslant x)$.
Show
$$\forall n \in \mathbb{N}^{\star}, \forall x \in D_n, \quad F_n(x) = x + \frac{1}{2^n}.$$