Closed and bounded convergent sequences and subsets

• Oct 22nd 2012, 10:34 AM
renolovexoxo
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?
• Oct 22nd 2012, 11:04 AM
Plato
Re: Closed and bounded convergent sequences and subsets
Quote:

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