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

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- Apr 11th 2008, 08:26 AMjames_bondThere is a rational and an irrational number in any interval
Prove that in any interval there is a rational and an irrational number.

- Apr 11th 2008, 11:35 AMAryth
Dr. Math: Go here

This is a proof of just what you are asking. - Apr 11th 2008, 12:16 PMPlato
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 . - Apr 28th 2008, 05:02 PMJaneBennet
- Apr 29th 2008, 01:06 AMIsomorphism
- Apr 30th 2008, 06:17 AMJaneBennet