We present a detailed and elementary construction of the real numbers from
the rational numbers a la Bourbaki. The real numbers are defined to be the set
of all minimal Cauchy filters in $\mathbb{Q}$ (where the Cauchy condition is
defined in terms of the absolute value function on $\mathbb{Q}$) and are proven
directly, without employing any of the techniques of uniform spaces, to form a
complete ordered field. The construction can be seen as a variant of Bachmann's
construction by means of nested rational intervals, allowing for a canonical
choice of representatives