We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Conversely, let $n$ be a non-zero natural number and let $X_n$ be a random variable that follows a uniform distribution on $D_n$. Show that there exist random variables $V_1, \ldots, V_n$ mutually independent, each following a Bernoulli distribution with parameter $1/2$, and such that
$$X_n = \sum_{k=1}^{n} \frac{V_k}{2^k}.$$