What is a simple proof that the set of normal numbers in base 2 in $[0,1]$ is of the first category (meager)?
Definition 1. A set is called meager or of the first category if it is the countable union of nowhere dense sets.
Definition 2. A number $x=.x_1x_2x_3...$ written in its binary expansion is called normal in base 2 if $$ \lim_{n \to \infty} \frac{1}{n}\sum_{i=1}^nx_i = \frac{1}{2} $$
For any natural number $n$, there is a dense open set $G_n$ of points $x\in[0,1]$ satisfying the following condition: For some $k\geq n$, the binary expansion of $x$ has only 0's from position $k$ to position $2^k$. ($G_n$ doesn't contain all such points $x$; some endpoints have to be removed to make $G_n$ open.) The complement of $G_n$ is then nowhere dense, and so the complement of $\bigcap_nG_n$ is of first category. But that complement contains all normal numbers in base 2.