A number xx is said to be
an accumulation point of a non-empty set AâRAâR 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.