Let $n$ be a non-zero natural number. Show by induction
$$\forall k \in \llbracket 0, 2^n - 1 \rrbracket, \quad k \in \operatorname{Im} \Phi_n$$
where $\Phi_n : \{0,1\}^n \rightarrow \llbracket 0, 2^n - 1 \rrbracket$, $(x_j)_{j \in \llbracket 1,n \rrbracket} \mapsto \sum_{j=1}^{n} x_j 2^{n-j}$.