We denote
$$D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\} \quad \text{and} \quad D = \bigcup_{n \in \mathbb{N}^{\star}} D_n.$$
Establish the monotonicity in the sense of inclusion of the sequence $(D_n)_{n \geqslant 1}$ then verify $D \subset [0,1[$.