As I said, that was really an illustration, not a proof. The proof is that if we assume 1-0.999... is non-zero, we arrive at a contradiction. I stress that no "infinitesimals" are involved. The proof only uses properties of real numbers.This shows only that decimal representation is only an approximation of fractionsie, 0.333... is not exactly 1/3, but infinitesimally shy of 1/3. Thus, we just don't worry about it and define that 0.333... is 1/3, and 0.999... is 1. It'll take a while for the error to cause problems
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(My illustration really assumes what it wants to prove. Once we accept that 1/3=0.333... , multiplying by both sides gives 1=0.999... It's just that, to most people 1/3=0.333... is more intuitive than 1=0.999... - but if one is true, so must be the other)