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.
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.