What's the Point That Rudin is Trying to Make After Theorem 3.29? After proving Theorem 3.29 in page 63 of Rudin's "Principles of Mathematical Analysis", Rudin says that "this procedure may evidently be continued", what procedure is he talking about exactly? Moreover, I don't really understand the point he is trying to make after that before going to define the number $e$. It would really be helpful for me if someone could explain more in detail and intuitively what he is trying to say.


 A: The idea is this: We are discussing series $\sum_{k=0}^\infty a_k$ with  terms $a_k>0$. Some of these series converge, some diverge. Where is the boundary?
Hereabout one can prove the following facts (and some of this is contained in Rudin's Principles):
If the series $\sum_{k=0}^\infty a_k$ with $a_k>0$ is convergent you can construct a series $\sum_{k=0}^\infty b_k$ with $$\lim_{k\to\infty}{b_k\over a_k}=\infty\ ,$$
which is still convergent. Conversely: If $\sum_{k=0}^\infty a_k$ is divergent you can construct a series $\sum_{k=0}^\infty b_k$ with $$\lim_{k\to\infty}{b_k\over a_k}=0\ ,$$
which is still divergent.
A: The procedure in question is that used in the proof to deduce the convergence of the series $\sum \frac{1}{n(\log n)^p},$ and from the next sentence he means the case when you apply this procedure to monotonically decreasing series with smaller terms, specifically those of the form $$\sum \frac{1}{n(\log n)^p(\log \log n)^p(\log\log\log n)^p\cdots(\log\log\log\log\cdots\log n)^p}.$$
What he's saying in the next paragraph is that if you consider series of this form, for increasingly many iterations of the logarithm function, then you would always discover that we always have the series divergent for $p\le 1$ and convergent otherwise. However, when you take two values of $p$ that are as close to each other as you wish, but so that one is $\le 1$ and the other is $>1,$ then the corresponding series differ very little from each other, yet one always diverges and the other always converges. His claim is now that one might intuitively expect that there would be some sort of separation, a cut as it were, in the values of $p,$ so that all convergent series of this type are in one part and all divergent ones are in the other part. Now the stupendous line: Rudin claims that no matter how we make this concept of cut precise, no such separation can exist.
Of course all these claims are unsubstantiated, but he refers his reader to Knopp.
