A number xx is said to be
an accumulation point of a non-empty set <span>A⊆R</span><span>A<span>⊆R if every
neighborhood of xx contains at
least one member of AA which is
different from xx.</span></span>
A neighborhood of xx is any open
interval which contains xx.
<span><span>In this question, we have <span><span>A=Q</span><span>A=Q</span></span></span> and
we need to show if <span>xx</span> is
any real number then <span>xx</span> is
an accumulation point of <span>QQ</span>.
This is almost obvious because if <span>xx</span> is
any specific real number then any neighborhood <span>BB</span> of <span>xx</span> contains
infinitely many rational numbers (and hence at least one of them is different
from <span>xx</span> itself).</span>
The fundamental property which we are using here is the following:
If <span>a<b</span><span>a<b</span> are two real
numbers then there is a rational xx with <span>a<x<b</span><span>a<x<b</span> and an
irrational number yy with <span><span>a<y<</span>b</span><span>a<y<b</span>.
<span>
</span>
<span><span>This above fact implies that there are infinitely many rational
and irrational numbers between <span>aa</span>
</span>and <span>bb</span>.
In other words any interval <span><span>(a,b)</span><span>(a,b)</span></span> contains
infinitely many rational and irrational numbers. The neighborhood <span>BB</span> in
my answer above is an interval of this type and hence contains many rational
numbers.</span>
