This is a question about the logic of Theorems 2.41, 2.42 in Rudin's 3rd Ed, which deal with the Heine-Borel and Weierstrass properties of sets of $R^k$, respectively.
A quick version of my question is: doesn't 2.41 (together with 2.40, on which it depends) moot 2.42?
2.41. (Heine-Borel+) If a set E in $R^k$ has one of these three properties, it has the other two: (a) E is closed and bounded; (b) E is compact; (c) Every infinite subset of E has a limit point in E.
2.42. (Weierstrass) Every bounded infinite subset of $R^k$ has a limit point in $R^k$.
The text explains that 2.41(a) implies (b) implies (c) implies (a). So the only step to be supplied for 2.42 is that a bounded subset of $R^k$ is compact (and therefore closed, to bring it into the ambit of 2.41).
But in both theorems Rudin resorts to an extrinsic proof (his 2.40) to show that that "k-cells" are compact.
I think (am not sure) there is a theorem stating that Heine-Borel implies Weierstrass (and conversely) but H-B consists of 2.41(a) and (b), according to Rudin's note preceding 2.41. I wonder if, with the addition of 2.41(c), he needs to prove Weierstrass separately?
This is a sort of fussy question but an answer might help me understand the relationship between these ideas better. Thanks. EDITED: so the quick version of the question includes the ref. to 2.40.
Best Answer
The connection between the topics is strong, but not as strong as you currently believe. The problem is that a bounded set of $\mathbb{R}^k$ is not necessarily compact: take the open unitl ball in $\mathbb{R}^k$, or more generally any bounded set in $\mathbb{R}^k$ that is not closed. The k-cells are invoked to say that our closed and bounded set $E$ is a subset of some k-cell $I$, as it is bounded. It would be enough to take $E$'s closure, but this approach also works: as k-cells are compact, 2.41 tells us that every infinite subset of $I$ (which includes the infinite subsets of $E$) has a limit point in $I \subset \mathbb{R}^k$ (and not necessarily $E$).
Further examining Rudin's setup, both the given proofs for Heine-Borel and Weierstrass use the general observations that infinite subsets of compact sets have a limit point in that compact set, as well as the compactness of k-cells and that closed sets of compact sets are compact.