Baire category theorem for $\mathbb{R}$

Statement (Baire). $\mathbb{R}$ is not a countable union of nowhere dense sets.

Proof. Suppose $\mathbb{R}=\bigcup_n F_n$ with each $F_n$ nowhere dense. Construct nested closed intervals $I_n$ with $I_n\cap F_n=\emptyset$ and $\mathrm{diam}(I_n)\le2^{-n}$. Their intersection is a point $x\notin\bigcup F_n$ — contradiction.

Conclusion. $\mathbb{R}$ is of second Baire category, in particular uncountable.

Remark. A dense $G_\delta$ (e.g. irrationals) is residual; its complement is meager.