Motivation for Baby Rudin Theorems 2.38-2.40 (Compactness, k-cells)

Question

I would appreciate some context around Baby Rudin's Theorems 2.38-2.40. It's in the section dealing with compactness. I find hard to give any motivations to these theorems in particular. Why are they important? Why are these theorems selected and not others?

These are said theorems:

\begin{array}{l}\text { 2.38 Theorem. If }\left\{I_{n}\right\} \text { is a sequence of intervals in } R^{1} \text { , such that } I_{n} \supset I_{n+1} \\ (n=1,2,3, \ldots), \text { then } \bigcap_{1}^{\infty} I_{n} \text { is not empty. }\end{array}

and

\begin{array}{l}\text { 2.39 Theorem. Let } k \text { be a positive integer. If }\left\{I_{n}\right\} \text { is a sequence of } k \text { -cells such } \\ \text { that } I_{n}\supset I_{n+1}(n=1,2,3, \ldots), \text { then } \bigcap_{1}^{\infty} I_{n} \text { is not empty. }\end{array}

and

\begin{equation} \text { 2.40 Theorem. Every k-cell is compact. } \end{equation}

In particular, no other book that I know uses the concept of k-cell so it is hard to imagine why are proofs involving it important.

Thank you!

Answer 1

Note: When studying Rudin page-by-page without assuming any other theory, you do not learn that intervals in $\Bbb R$ are compact until studying the proof of

\begin{array}{l}\text { 2.40 Theorem. Every } k \text{-cell is compact.}\end{array}

Moreover, writing in a terse style, Rudin doesn't even remark that

$\quad$ ... every interval subset of $\Bbb R$ is compact

---

To gain an appreciation (or not) of Rudin's development the OP is encouraged to use other resources to formulate proofs of these two results:

It might have been instructive if Rudin had used the word proposition instead of theorem for some results concerning $k\text{-cells}$, so that upon reaching, say,

\begin{array}{l}\text { 2.39 Proposition. Let } k \text { be a positive integer. If }\left\{I_{n}\right\} \text { is a sequence of } k \text { -cells such } \\ \text { that } I_{n}\supset I_{n+1}(n=1,2,3, \ldots), \text { then } \bigcap_{1}^{\infty} I_{n} \text { is not empty. }\end{array}

we would be 'on notice' that this result is ancillary to the theorems.

Upon skimming his book it appears that he never uses the word lemma to highlight results.