Infinite sets, intersection, nested intervals.

[mathkiddi]

Homework Statement

Let [a,b] be an interval and let A be a subset of [a,b]. and Suppose that A is an infinite set.

Let z be the unique point that belongs to all of the intervals [a_n, b_n]. Show that if I is any interval that contains z, then A intersect I is infinite.

Homework Equations

I don't know where to start this problem but I think I can use the fact that a set that contains an infinite subset is infinite. Any help would be appreciate.

The Attempt at a Solution

I know I is an infinite interval, I think. I also know the set of intervals [a_n, b_n] intersect A is infinite. I believe I can say that [a_n, b_b] is a subset of [a,b] and that [a,b] is infinite.

 

[kurt.physics]

I know that z is the unique point that belongs to all of the intervals [an, bn]. So I think I can say that [an, bn] intersect A is infinite, which means [a,b] intersect A is infinite. I then can say that I intersect A is infinite because I contains z, which belongs to all of the intervals [an, bn].