We're given n surjections fn= : {1,2,...,N} -> En. for all n in the Natural Numbers.

We have to prove that E= the union of all En is countable by defining a surjection from the natural numbers to the union of sets.

Since all the En have at most N elements, could I just define my surjection from {1,2,...,nN} and my function would be

F(i)={i for i in E

{s0 for i not in E

where so is a fixed element in E

E here could cover all of the Natural Numbers.