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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?

Re: Closed and bounded convergent sequences and subsets

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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 $\displaystyle F~.$