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?