By F. Hirsch, G. Mokobodzki

**Read or Download Seminaire de Theorie du Potentiel Paris No 3 PDF**

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B maps A onto B if to each element b of B there corresponds an element a of A with f(a) = b. In other words, f maps A onto B if there is an operation g from B into A such that f(g(b)) = b for each b in B. A set A is countable if there exists a mapping of lL+ onto A, intuitively, this means that the elements of A can be arranged in a sequence with possible duplications. The elements of the cartesian product lL x lL of the set lL of integers with itself can be arranged in a sequence as follows. We order the elements (m, n) of lL x lL, first according to the value of Iml + Inl, then according to the value of m, and finally according to the value of 1 Sets and Functions 17 n.

19) Theorem. Let (an) be a sequence of real numbers, and let Xo and Yo be real numbers with Xo < Yo Then there exists a real number x such that xo~x~Yo and X =1= an for all n in 7l+. Proof: We construct by induction sequences (xn) and (Yn) of rational numbers such that (i) xO~xn~xm