Thread: Closed and bounded convergent sequences and subsets

1. 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?

2. Re: Closed and bounded convergent sequences and subsets

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 $\mathcal{O}=F^c$, the complement of $F$.
The complement of a closed set is an open set.
If $p\notin F$ then $p\in\mathcal{O}$.
Thus $p$ belongs to an open set that contains no point of $F$.
That contradicts the given that $p$ is a limit point of $F~.$