In this paper we will give a proof of Kolmogorov's theorem on
the conservation of invariant tori. This proof is close to
the one given by Bennettin, Galgani, Giorgilli and Strelcyn
in ; we follow the outline of their proof, but carry out the steps
somewhat differently in several places. In particular, the use of balls rather
than polydiscs simplifies several arguments and improves the estimates.