The main purpose of this note is to give a simplified and elementary proof of Emile Borel’s well-known result on the normality of real numbers published in 1909. He proved that Lebesgue almost every real number \(\omega\) is normal, that is, \[ \lim_{n\to\infty} \frac{a_1(\omega)+\cdots+ a_n(\omega)}{n}= \frac{1}{2}, \] where \(a_j(\omega)\in \{0,1\}\) is the \(j\)th digit in the binary expansion of \(\omega\in [0,1)\). The author must confess that this proof cannot go through without using arguments from measure theory, since the notion of “a set \(N\) of Lebesgue measure zero” is intrinsically needed. To overcome this difficulty, the following quite accessible definition of \(N\) is used: For each \(\varepsilon> 0\) there is a sequence of intervals \((\alpha_n,\beta_n)\), \(n=1,2\dots\), such that \[ N\subseteq \bigcup_{n=1}^\infty (\alpha_n,\beta_n) \quad\text{and}\quad \sum_{n=1}^\infty (\beta_n-\alpha_n)< \varepsilon. \]