Dr. Math: Go here
This is a proof of just what you are asking.
Dr. Math: Go here
This is a proof of just what you are asking.
Lemma: If then .
Proof:
By the floor function, .
Between any two numbers there is a rational number.
Proof:
If then .
But that means that so by the lemma .
Between any two numbers there is a irrational number.
Proof:
If then so by the above .
But and is irrational .