Is size N (infinite) union of countable sets at most as big as N?

Question

Say I have a set S of an arbitrary infinite size N, and each member of S is a countable set. Is the union always at most size N? I know that Tarski has a theorem that AxA is equinumerous to A for any infinite set A, and that this is equivalent to the axiom of choice, and this seems like it could prove what I want, as the union of N countable sets should inject to SxS. Is this valid? And is there a way to prove this without the axiom of choice?