Let [a,b] be an interval and let A be a subset of [a,b] and suppose that A is an infinite set.
Suppose that A is uncountable. Prove that there exists a point z which is an element of [a,b] such that A intersect I is uncountable for every open interval I that contains z.
I don't really know how to start this problem. I know I can use the fact that a set that contains an uncountable subset is uncountable. Any help would be appreciated
The Attempt at a Solution
I know z is an element of I, and that I is uncountable. I also know z is an element of [an, bn]. I know there exists a unique point z in [an, bn] and I know A is uncountable.