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Problem 5 (due on Monday, November 7)
Let F be the set of all functions f:R⟶N from the real numbers to natural numbers. Prove that there exists a sequence of functions f1,f2,f3,… in F such that for every finite set A⊆R and every g∈F there is i such that g(a)=fi(a) for every a∈A.
The problem was solved by Levi Axelrod and Ashton Keith. The solutions differ in details but follow similar idea. For a detailed solution and some applications to topology see the following link Solution.