$\mathbb{R}$ is uncountable:
The set $\{0,1\}^\mathbb{N}$ is uncountable using Cantor's diagonal argument: if it were countable, list all the sequences and take the diagonal, it is a sequence which is not on the list, which is absurd. Then the open interval $(0,1)$ is uncountable via (taking the infinite) binary expressions: the result follows because $\mathbb{R}$ is in bijection with $(0,1)$ via $x\mapsto \frac{2}{\pi}\tan(x)$, for example.