Resonant and non-resonant tori density in non-degenerate system

I’m following the discussion on the page 290 of Mathematical Methods of Classical Mechanics by V. I. Arnol’d (you can download it here), and I’ve encountered the fact that in a nondegenerate system, the resonant and nonresonant tori are dense – even if the former have Lebesgue measure zero and the latter are of full measure.

Such fact is asserted in various sources around the web, but I couldn’t find a proof – in fact, could you help me in answering the two problems proposed by Arnol’d?