  There are many ways to construct the field R of real numbers. The most
important and famous of these employ Cauchy sequences (Cantor) or cuts
(Dedekind) in the field Q of rational numbers. These constructions sometimes
overlook important details and often are complicated and cumbersome. In this
note, the authors propose an essential, clear and rigorous construction of R
from the stucture Q+ of positive rational numbers by the key notion of initial
segment.
