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I am reading "Principles of Mathematical Analysis 3rd Edition" by Walter Rudin.

In Remark 4.31 on p.97, Rudin wrote this symbol $$\sum_{x_n < x} c_n.$$

What is the definition of this symbol rigorously?

I guess this symbol is equivalent to $$\sum_{n \in \{i | x_i < x\}} c_n.$$

Let $S$ be a subset of $\mathbb{N}$.

In general, what is the definition of the following symbol? : $$\sum_{n \in S} a_n.$$

If $S$ is finite, then the definition of $$\sum_{n \in S} a_n$$ is clear.

So let's consider the case in which $S$ is infinite.

Since $S$ is countable, there is a bijection $\phi : \mathbb{N} \to S$.

Is $$\sum_{n \in S} a_n := \sum_{i \in \mathbb{N}} a_{\phi(i)}$$ the definition of $\sum_{n \in S} a_n$?

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    $\begingroup$ Your interpretation is right $\endgroup$ – YuiTo Cheng Feb 13 at 4:40
  • $\begingroup$ @YuiToCheng Thank you very much. $\endgroup$ – tchappy ha Feb 13 at 4:42
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    $\begingroup$ it should be noted that the sum is possibly dependent on the choice of bijection if the $a_i$'s are not all non-negative $\endgroup$ – mathworker21 Feb 13 at 4:44
  • $\begingroup$ @mathworker21 In that case, I would write $\sum_{n=0}^{\infty}a_n$ to specify that bijection. $\endgroup$ – YuiTo Cheng Feb 13 at 4:48
  • $\begingroup$ @mathworker21 Thank you very much. $\endgroup$ – tchappy ha Feb 13 at 5:04

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