Proof that in any open interval there is infinitley many irrationals

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.

Re: Proof that in any open interval there is infinitley many irrationals

Quote:

Originally Posted by

**hmmmm** 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.

Here is one comment. Let

You have proven that any open interval contains an irrational.

Once you find then find .

If find .

Re: Proof that in any open interval there is infinitley many irrationals

Yeah thanks for the help there, thought that it was a bit poor at the end there thanks.

Re: Proof that in any open interval there is infinitley many irrationals

Quote:

Originally Posted by

**hmmmm** 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.

Another possibility would be to note that if there were finitely many irrationals in then would be countable...but I'm sure you this isn't true.