Examples of non-constructive results I'm giving a talk on constructive mathematics, and I'd like some snappy examples of weird things that happen in non-constructive math.  
For example, it would be great if there were some theorem claiming $\neg \forall x. \neg P(x)$, where no $x$ satisfying $P(x)$ were known or knowable.
But other examples are welcome as well.

If $\Phi$ is any statement, the following is a consequence of the law of the excluded middie:
$$
(\exists n \in \mathbb{N})[(n = 0 \land \Phi) \lor (n \not = 0 \land \lnot \Phi)]
$$
It will only be provable constructively if either $\Phi$ or $\lnot \Phi$ is provable constructively, because to prove it constructively you would have to produce an actual value of $n$, which means you would have to decide $\Phi$. 
