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.