(a) If $\{P_n\}$ is a sequence in a compact metric space $X$, then some subÂsequence of $\{P_n\}$ converges to a point of $X$.
(b) Every bounded sequence in $\mathbb R^k$ contains a convergent subsequence.
Rudin proves (a), then argues for (b) as follows:
"(b) This follows from (a), since Theorem 2.41 implies that every bounded subset of $\mathbb R^k$ lies in a compact subset of $\mathbb R^k$," where Theorem 2.41 is the Heine-Borel Theorem.
But doesn't this require the set to be bounded AND closed? From what I can see Rudin makes no argument about that the sequence in $\mathbb R^k$ is closed.
Best Answer
He is not saying that the bounded set itself is compact. What he is saying is that that bounded set is contained in some compact set. As a bounded set is, by definition, contained in an open ball, it is also contained in the closure of the ball, which is compact.