$\mathbb{R}$ is uncountable

Setup. Suppose $(0,1)$ is countable, with enumeration $x_1,x_2,\ldots$.

Diagonal. Write each $x_n$ as a decimal expansion avoiding trailing $9$'s. Define $y=0.d_1d_2\ldots$ by $d_n=1$ if the $n$-th digit of $x_n\neq1$, and $d_n=2$ otherwise.

Contradiction. $y\in(0,1)$ but $y\neq x_n$ for all $n$ (differ in $n$-th place). Contradicts enumeration.

Conclusion. $\mathbb{R}$ is uncountable.