True or false

Between any two unequal irrational numbers there is a rational number.

I'm leaning towards true, but I can't come up with a convincing argument.

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- Feb 11th 2009, 07:43 PMnoles2188Rational and irrational numbers
True or false

Between any two unequal irrational numbers there is a rational number.

I'm leaning towards true, but I can't come up with a convincing argument. - Feb 12th 2009, 02:42 AMclic-clac
Hi

What you think is right, what is said is that is dense in

Let be two (unequal) reals. Then Since the sequence strictly decreases and has limit there is a such that

Let be the greatest integer such that Then therefore

Since are integers, is a rationnal; we've proved that between two unequal reals lies a rational number. In particular, there's always a rational number between two unequal irrational numbers.