A real is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals. A real is random if its binary expansion is a random sequence. Our aim is to o er a self-contained proof, based on the papers (Calude et al., in: M. Morvan, C. Meinel, D. Krob (Eds.), Proc. 15th Symp. on Theoretical Aspects of Computer Science, Paris, Springer, Berlin, 1998, pp. 596