Maybe you want to say an irrational number because is a real number such that

Let be an irrational number. Then the sequence decreases and has for limit. So there is a such that

We just have to prove that is irrational: if that wasn't the case, since then would be a rational number, contradiction. Thus is an irrational number, and since there exist positive irrational numbers, that's ok.

An example:

At first sight, I would believe B

If A tells the truth, then he lies... Impossible

If C tells the truth (which implies that B lies), then B is right... Impossible

If B lies, then A tells the truth... Impossible