Is this a typo? Baby Rudin Theorem 3.27 is not proved correctly

Question

This has been referred to as the Cauchy condensation test.

He only shows the partial sum sequence is bounded not convergent. To do the latter, must show that for $n>N$ the partial sums are within $\epsilon$ of the limit for any $\epsilon > 0$. The sums have room to move around in between $S$ and $2S$ as stated in the book.

The post is not self-contained. Please consider adding the proof in the post. (FYI: There's another question on this Theorem: here) — Arctic Char, Oct 10 '20 at 20:04
Without having had a look at the book, is Rudin maybe implicitly using the fact that increasing sequences converge if they are bounded? — Bananach, Oct 10 '20 at 20:06
It is stated and proved correctly in the first edition. As noted there, it suffices to show boundedness, since the terms are non-negative. — Brian M. Scott, Oct 10 '20 at 20:06

score 1 · Answer 1 · answered Oct 10 '20 at 20:10

1

As stated in the helpful comments, boundness is implied by 3.24. Case closed. I spent over an hour after missing this little detail. thanks everyone.

answered Oct 10 '20 at 20:10

BigPaper

33

Thanks for your question, I missed this also :) – Shaq Xu Dec 23 '21 at 11:16

Is this a typo? Baby Rudin Theorem 3.27 is not proved correctly

1 Answers1