Thread: Closed and bounded convergent sequences and subsets

I've attached the problem. The last part of part a is: prove p is in F (in case it is hard to read)

To me it intuitively makes sense that this p would be contained in F when it is closed and bounded, but I am having a ton of trouble seeing how to start the formal proof. Am I supposed to be using compactness?

Originally Posted by renolovexoxo

I've attached the problem. The last part of part a is: prove p is in F (in case it is hard to read)
To me it intuitively makes sense that this p would be contained in F when it is closed and bounded, but I am having a ton of trouble seeing how to start the formal proof. Am I supposed to be using compactness?

The proof is trivial. Let , the complement of .
The complement of a closed set is an open set.
If then .
Thus belongs to an open set that contains no point of .
That contradicts the given that is a limit point of