Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Show that for every non-empty interval $I \subset [0,1]$, we have
$$\lim_{n \rightarrow \infty} \mathbb{P}(Y_n \in I) = \ell(I) \quad \text{with} \quad \ell(I) = \sup I - \inf I.$$