Proving Every Uncountable Subset on Real Line Has Limit Point | Hint Included

[qinglong.1397]

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]

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!

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]

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?

Yes. I can. It is pretty easy.

 

[Dick]

Yes. I can. It is pretty easy.

Well, then, if the set doesn't have a limit point, then every interval must contain a finite number of points, right?

 

[qinglong.1397]

Well, then, if the set doesn't have a limit point, then every interval must contain a finite number of points, right?

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]

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?

The real line is a countable union of bounded intervals. Intersect your set with each interval.

 

[qinglong.1397]

The real line is a countable union of bounded intervals. Intersect your set with each interval.

Oh, I see. Thank you very much!