Question

How to prove a number is an accumulation point of a set?

Answer 1

A number xx is said to be an accumulation point of a non-empty set A⊆RA⊆R if every neighborhood of xx contains at least one member of AA which is different from xx.

A neighborhood of xx is any open interval which contains xx.

In this question, we have A=QA=Q and we need to show if xx is any real number then xx is an accumulation point of QQ. This is almost obvious because if xx is any specific real number then any neighborhood BB of xx contains infinitely many rational numbers (and hence at least one of them is different from xx itself).

The fundamental property which we are using here is the following:

If a<ba<b are two real numbers then there is a rational xx with a<x<ba<x<b and an irrational number yy with a<y<ba<y<b.

This above fact implies that there are infinitely many rational and irrational numbers between aa and bb. In other words any interval (a,b)(a,b) contains infinitely many rational and irrational numbers. The neighborhood BB in my answer above is an interval of this type and hence contains many rational numbers.