Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set
$$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}, \quad F_n(x) = \mathbb{P}(Y_n \leqslant x), \quad G_n(x) = \mathbb{P}(Y_n < x).$$
Deduce the pointwise convergence of the sequences of functions $(F_n)_{n \geqslant 1}$ and $(G_n)_{n \geqslant 1}$.