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.