115 says the set of irrational numbers is dense in the ratio

1.15 says the set of irrational numbers is dense in the rational numbers

Suppose we believe the conclusion of Group Assignment 2. #2. which reads Let a be an irrational number. For any epsilon > 0, no matter how small, there is a rations number x in the interval (a - epsilon, a + epsilon). Let I = R\\Q be the set of irrational real numbers. Combine this result with Theorem 1.15 to prove. The irrationals I are dense in R

Solution

Solution: Q is dense in R, so Q+sqrt2 is dense in R+sqrt2=R. Since Q+sqrt2 is a subset of the irrationals, we conclude that the irrationals are also dense in R.

Or here is another way:

By the density of rational numbers, there exists a rational number r(x,y).

Since (yr)/2>0 by the Archimedian Property there exists nN such that (yr)/2>1/n. Then we have x<r+2/n<r+4/n<y. Now check that s=r+2/n is an irrational number sitting in (x,y).