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