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