I am following the proof of Cantor's theorem that every countable closed set is of uniqueness as given in Kechris' notes (available here, thm. 4.2, proof on p. 12).
My doubts are the following: how can he claim that the translation of a set of uniqueness is still of uniqueness? I mean, that is intuitively clear, but I don't see how can he claim that with the tools he developed up to that point. Moreover (and maybe more importantly) where does the assumption that $0\notin E$ comes into play? Why doesn't the same proof work if we assume that $0\in E$?