Steinhaus' theorem states that if A and B are sets of real numbers with positive Lebesgue measure, their sumset A + B has nonempty interior (i.e. includes an interval). This talk describes the result of applying a certain translation, heuristically, to a nonstandard proof by Renling Jin, to find an elementary proof of (a constructive version of) the theorem.
Jin's proof of Steinhaus' theorem is found in his paper "Sumset phenomenon." See the list of publications accessible from his home page.
Steinhaus originally stated the theorem in terms of distance sets (replace B by -B). The article appeared in 1920 in the first volume of Fundamenta Mathematicae. Remarkably, it is available online.