$\mathbb{R}$ represented using an infinite union of finite sets containing reals? If $S_1$, $S_2$, $\dots$ are sets of real numbers and if $\bigcup_{j=1}^{\infty}{S_j} = \mathbb{R}$ then one of the sets $S_j$ must have infinitely many elements.
I believe at least one of the $S_j$ must be an infinite set, but I can't work out a proof. What's the trick I'm missing? 

Suppose that all sets were finite, the union of countably many finite sets is countable, but the real numbers are not.
[This argument uses the axiom of choice, however it is true without the axiom of choice that a countable union of finite sets of real numbers is countable]
