How would you show that it is "impossible to construct a sequence such that ? Wouldn't that depend on exactly what E was?

I would do this: let [tex]E_N[tex] be the set of all sequences such that but that for all n> N. It should be easy to show that [tex]E_N[tex] is countable. And E is the union of all [tex]E_N[tex] for all positive integers, N. That is, E is the countable union of countable sets.