The paper [5] contains a very nice exposition of the classical KAM theorem which has been very influential. It is our purpose in this short and non-self-contained note to add two remarks to this remarkable paper.

The first one concerns a technical mistake in the proof of the main abstract statement Theorem A, 1 which has been recently pointed out and corrected in the PhD thesis [3]. Yet a correction of this mistake, following Pöschel arguments, leads to a final statement which is both less elegant and quantitatively weaker. We would like to explain how, by modifying slightly the arguments using ideas due to Rüssmann (see for instance [7]), Theorem A of [5] can be proved without any changes. The aforementioned modifications consist of replacing the crude Fourier truncation by a more refined polynomial approximation, and then set an iterative scheme with a linear,2 rather than super-linear, speed of convergence