PROOF. Suppose that there is a 1-1 function f that takes N onto the interval

(0,1). Write the numbers f(j), j E N, in decimal notation, using the finite expansion when possible; i.e., f(l) = 0.an a 12""

f(2) = O.a21 a 22···, f(3) = 0.a31 a 32 ... ,

... ,