This is a two-part question (if that's ok!):

Given a collection of pairwise disjoint countable sets {A_n : n N}, prove that the unionA=nNAn is also countable by defining a bijection betweenNN and A in two ways:

(1) Assume for each n, there is a bijection f_n : N A_n, and define the assignment F :NN A by F(n, m) = f_n(m). How do I prove F is a function, that is one-to-one and onto?

(2) If I am again assuming for each n, there is a bijection g_n : A_n N, how do I now prove that the assignment given below is a function, and is one-to-one and onto?:

G : A NN defined by for each x A, G(x) = (n, g_n(x)) (where n is the index of the set where x A_n)