Using the results of the previous questions (in particular that $\Lambda$ is a bijection from $\{0,1\}^{\mathbb{N}}$ to $[0,1[$, and that $\mathcal{P}(\mathbb{N})$ is not countable), conclude that $[0,1[$ is not countable.
Using the results of the previous questions (in particular that $\Lambda$ is a bijection from $\{0,1\}^{\mathbb{N}}$ to $[0,1[$, and that $\mathcal{P}(\mathbb{N})$ is not countable), conclude that $[0,1[$ is not countable.