[13] A real number r [0, 1] is called a computable real number if there exists computable function such that r = 0.

[13] A real number r ∈ [0, 1] is called a computable real number if there exists computable function φ such that r = 0.ω with φ(i) = ωi, for all i. We call ω a computable sequence. A computable sequence of computable reals is a sequence r1, r2,... of reals if there is a computable function ψ in two arguments such that ψ(i, j) = ri,j and ri = 0.ri,1ri,2 ... .

Show that not all real numbers are computable; show that there are only countably many computable numbers; and show that there is a computable sequence of computable reals that converges to a real, but not to a computable one.