The set of irrational numbers between 9 and 10 is countable.
The Attempt at a Solution
My belief is that I can prove by contradiction.
first, i must prove by contradiction using diagonalization that the real numbers between 9 and 10 are uncountable. (1)
second, i take set of rational numbers Q is countable, hence a subset Q (9,10) is also countable. (2)
third i can prove by contradiction stating I (9, 10) is countable
Q(9,10) [Countable per item (2) ] U I(9,10) [ Countable per statement]
This would imply that by closure properties R(9,10) is countable. which is a condtradiction of what we found in 1.
Is this logic sound? Can I prove (1) using the same diagonalization method used for (0,1)?