Irrationals vs rationals

Question

In real number system, given a number"n" whats the number immediately after"n" lets say the number after "n" is "y". Is "y" rational or irrational??? By the completeness property of real number system "y" can be both rational and irrational.. So "y" does not exist?

Answer 1

In the real number system, there is no number immediately after $n$. Such a number would be the least number greater than $n$--that is, the least element of $(n,+\infty),$ which has no least element.

The standard ordering of the reals is a dense linear order, meaning that between any two distinct elements, there is another element. Thus, there can be no least number greater than $n$, since if $y$ is a number greater than $n,$ there is some $n<z<y$.