It is not known if every finitary factor of a Bernoulli scheme is finitarily isomorphic to a Bernoulli scheme (is finitarily Bernoulli). In this paper, for any Bernoulli scheme X, we define a metric on the finitary factor maps from X. We show that for any finitary map f: X → Y, there exists a sequence of finitary maps fₙ: X → Y(n) that converges to f, where each Y(n) is finitarily Bernoulli. Thus, the maps to finitarily Bernoulli factors are dense. Let (X(n)) be a sequence of Bernoulli schemes such that each Y(n) is finitarily isomorphic to X(n). Let X' be a Bernoulli scheme with the same entropy as Y. Then we also show that (X(n)) can be chosen so that there exists a sequence of finitary maps to the X(n) that converges to a finitary map to X'.