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In Rudin's Principles of Mathematical Analysis, there's a theorem that reads:

$\sum a_n$ converges if and only if for every $\epsilon > 0$, there is an integer $N$ such that $\lvert \sum_{k = n}^m a_k \rvert \le \epsilon$ if $m \ge n \ge N$. (3.22)

I'm having a bit of trouble understanding this theorem. Is it saying that every sum of $a_k$ from one point to another (both greater than or equal to $N$) must be less than some arbitrary epsilon? If so, what exactly is it "saying" and what's the significance? If someone could provide me with a simplified/"dumbed down" explanation of this theorem, it'd be tremendously helpful. Thank you.

Max
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    Are you familiar with the Cauchy criterion for convergence of sequences? Because this is just the Cauchy criterion applied to the sequence of partial sums of $\sum a_n$. – carmichael561 Oct 25 '17 at 16:16
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    @carmichael561 which of course Rudin says. – zhw. Oct 25 '17 at 16:38
  • In the first place it would be tremendously helpful if you would try to understand this theorem in the given formulation, and would not hope for a "dumbed down" version. Realizing that "segment sums" of a series are just differences of partial sums is a fundamental mathematical accomplishment. – Christian Blatter Oct 25 '17 at 19:06
  • This is just a limit definition, in the rigorous delta-epsilon fashion, being applied to a summation. – CogitoErgoCogitoSum Oct 26 '17 at 01:08

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