Show that the application $$\Psi : \begin{aligned} \{0,1\}^{\mathbb{N}} &\rightarrow [0,1] \\ (x_n) &\mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}} \end{aligned}$$ is well-defined and surjective. Is it injective?
Show that the application
$$\Psi : \begin{aligned} \{0,1\}^{\mathbb{N}} &\rightarrow [0,1] \\ (x_n) &\mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}} \end{aligned}$$
is well-defined and surjective. Is it injective?