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.
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