Let $E$ be a countable subset of $(a, b)$.
Let $\phi$ be a bijection from $\mathbb{N}$ to $E$.
Let $\{x_n\}$ be a sequence such that $x_n := \phi(n)$.
Let $\{c_n\}$ be a sequence of positive numbers such that $\sum c_n$ converges.
Let $f(x)$ be a function on $(a, b)$ defined by $f(x) := \sum\limits_{x_n < x} c_n$.
Does the following equality hold?:
$$f(x_n^+) = \inf_{x_n < t < b} f(t) = \sum_{x_i \leq x_n} c_i.$$
I am not sure that the above equality holds or not.
My attempt is here:
Since
$$\{i | x_i < x_n\} \cup \{n\} = \{i | x_i \leq x_n\},$$
$$f(x_n) + c_n = \sum_{i \in \{i | x_i < x_n\}} c_i + \sum_{i \in \{n\}} c_i = \sum_{i \in \{i | x_i < x_n\} \cup \{n\}} c_i = \sum_{\{i | x_i \leq x_n\}} c_i = \sum_{x_i \leq x_n} c_i.$$
If $t > x_n$, $$f(x_n) + c_n \leq f(t),$$ since $$n \not \in \{i | x_i < x_n\} \subsetneqq \{i | x_i < t\} \ni n.$$
