Is this proof ok?
If we take the open interval and shift it . Where where p is a prime.
Then we have that as the rationals are dense in the reals ( we can have a rational arbitrarly close to any real) we have:
Therefore as and we have
we have an irrational in any open interval.
Now as there are infinitely many primes we have infinitley many irrationals of the form , so we have infinitley many irrationals in the interval.
thanks for any help.