Clarification of a proof of completeness in classes of approximable optimization problems

We denote the real numbers by R and the positive real numbers by R+. Similarly we denote the rational numbers by Q and the positive rational numbers by Q+. We denote the natural numbers (including 0) by N. For all a and b in R with a < b, we denote the open interval between a and b by (a, b) and the closed interval by [a, b]. If S is a set, S∗ denotes the set of all finite sequences consisting of elements of S. For each finite sequence s, we denote the length of s by |s|. We denote the set of all finite binary strings by Σ∗.