$\mathbb{R}$ 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? 

In fact something stronger is true: at least one of $S_i$ has to be uncountable.
It is a standard fact that a countable union of countable sets is countable. You could for instance enumerate elements of $S_i$ so that:
$$ S_i = \{ S_i^{j} \}_{j \in \mathbb{N} } $$
and then arrange $\{ S_i^{j}: i,j \in \mathbb{N} \} = \bigcup_{i} S_i$ ordering first according to $i+j$ and then $i$. Since $\mathbb{R}$ is not countable, it can't be that all $S_i$ are countable, and you are done.
