Prove that in any interval there is a rational and an irrational number.
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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 .