Baby Rudin Theorem 2.41 boundedness

Question

I'm currently reading the POMA of Rudin and I don't understand the proof of boundedness of the theorem 2.41 (p. 40) of the book.

It wants to prove that if $E \subset \mathbb{R}^k$ we have that every infinite subset of $E$ has a limit point in $E$ implies that $E$ is bounded and closed. I have some troubles in understanding what the set $S$ is...

Is $S$ the $\mathbb{R}^k$ complement of the open ball centered around 0 of radius 1? If someone could explain me its proof this would be very nice,

Thanks

Answer 1

Because $E$ is unbounded, for each positive integer $n$ there is an $x_n\in E$ such that $|x_n|>n$.

$$S=\{x_1,x_2,\ldots,x_n,\ldots\}.$$

Does this clear things up?