The problem involves proving that every uncountable subset of the real line has a limit point. It references a hint regarding the union of a countable family of finite sets being countable.
The discussion is active, with participants offering hints and exploring various interpretations of the problem. There is a recognition of the relationship between uncountable sets and intervals, and some guidance has been provided regarding the intersection of sets with bounded intervals.
Participants are considering the implications of the problem within the constraints of the real line and the properties of bounded intervals. The discussion reflects uncertainty about the definitions and assumptions related to limit points and uncountable sets.
qinglong.1397 said:Homework Statement
Prove that every uncountable subset of the real lline has a limit point. There is a hint: The union of a countable family of finite sets is countable.
The Attempt at a Solution
Please give me more hints. Thanks a lot!
Dick said:If a closed bounded interval contains an infinite number of points, then it contains a limit point. Do you know this or can you prove it?
qinglong.1397 said:Yes. I can. It is pretty easy.
Dick said:Well, then, if the set doesn't have a limit point, then every interval must contain a finite number of points, right?
qinglong.1397 said:I think it is right for the bounded interval. But now, the interval is actually a set of some finite discrete points, isn't it?
So I should first suppose that the opposite statement is true. Then show that every uncountable subset of real line can be decomposed into countably many subsets. Some of them are intervals. Then I should prove that any interval has a limit point. So the uncountable subset consists of countable subsets which are finite. Is that right?
Dick said:The real line is a countable union of bounded intervals. Intersect your set with each interval.