Let be given with . Prove that there exists an irrational number such that .
Note: To do this problem, I think we need to assume the existence of an irrational number. We will prove the existence of such a number (the ) in I.3.12, Exercise #12.
Proof. Since the rationals are dense in the reals I.3.12, Exercise #6, we know that for with there exist such that
Now, assume the existence of an irrational number, say (see note preceding the proof about this). Since we know and from the order axioms exactly one of or is positive ( is nonzero since ). Without loss of generality, let . Then, since , we know there exists an integer such that
Also, since , we have ; thus, .
Then, by I.3.12, Exercise #7 we have irrational and hence irrational.
Thus, letting , we have with irrational