I'm working through Rudin right now, and I am stuck on one of his proofs. In theorem 2.41 he tries to prove that, if every infinite subset of a set $E$ contains a limit point in $E$, then $E$ is bounded.
To do prove this by contradiction, he attempts to construct an unbounded sequence $\{x_n\}$ such that $x_n \in E$ and $|x_n| > n$. Let $S$ denote the set of all $x_n$, which is clearly a subset of $E$.
While I understand why this set would be infinite, Rudin then asserts that this set has no limit point in $R^k$, which would be a contradiction.
I didn't really understand why this set has no limit point. Could someone help me out here?
I initially thought to prove that if there is some limit point $p \in R^k$, you would show that any neighborhood of $p$ could only contain a finite number of elements of $S$. But my proof ended up assuming that S is monotonically increasing, which is clearly not a general assumption.
Best Answer
Hint: Suppose $x_n\to x$, then $|x|\leq k$ where $k \in \mathbb{N}$. Can you solve it now?