# About $f(x) = \sum_{x_n < x} c_n$ in Remark 4.31 on p.97 in “Principles of Mathematical Analysis 3rd Edition” by Walter Rudin.

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 $$\sum_{n \in S} a_n$$?

• Your interpretation is right – YuiTo Cheng Feb 13 at 4:40
• @YuiToCheng Thank you very much. – tchappy ha Feb 13 at 4:42
• 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 – mathworker21 Feb 13 at 4:44
• @mathworker21 In that case, I would write $\sum_{n=0}^{\infty}a_n$ to specify that bijection. – YuiTo Cheng Feb 13 at 4:48
• @mathworker21 Thank you very much. – tchappy ha Feb 13 at 5:04