Answer to Question #135893 in Math for Inshal

Question

Find all limit points of the set of rational numbers.

Answer 1

To find the set of limit point we have to understand definition of limit point,

Limit point:a point for which every nbhd(neighborhood) contains at least one point belongs to a given set.

Now let talks about limit point of Q(set of rational number)

Take x be any arbitrary real number,

To prove that x is limit point Q,

We must have δ>0,(x-δ,x+δ) nbhd contains contains some point of Q other than x.

By densness property we can say that (x-δ,x+δ) contains infinitely many rational number [reason:Q is dense in R]

So x is limit point for Q.

But we take arbitrary x from R by that for all x in R is limit point for Q.

So the set of limit point of Q is R(set of real number).