# Real Analysis - Prove a set is countable

I am asked to prove the following question:

For each finite set of real numbers, $$F$$, define $$\sum F$$ to be the sum of all the numbers in $$F$$.

For each set $$P$$ of positive real numbers, define $$\sum P=\sup\{\sum F: F\text{ is a finite subset of } P\}$$.

Suppose $$P$$ is a set of positive real numbers such that $$\sum P < \infty$$. Prove that $$P$$ is a countable set.

My attempt (just a sketch proof):

Suppose $$P$$ is uncountable, $$\exists \epsilon_1 > 0$$, s.t. $$P_1 = [\inf P + \epsilon_1, \sup P]\cap P$$ is uncountable. Pick $$a_1 \in [\inf P, \inf P + \epsilon_1] \cap P \in P$$, such a $$P$$ must exists due to greatest lower bound property.

Now $$P_1$$ is uncountable, $$\exists \epsilon_2 > 0$$, s.t. $$P_2 = [\inf P_1 + \epsilon_2, \sup P_1]\cap P_1$$ is uncountable. Pick $$a_2 \in [\inf P_1, \inf P_1 + \epsilon_2]\cap P_1 \in P$$. Carry on this process (this should be formalized using induction), we can find $$a_1 \leq a_2 \leq a_3 \leq ...$$ with $$a_n > 0$$ and $$a_n \in P$$ for all $$n$$.

Now $$\sup\{\sum F_n: F_n = \{a_1,...,a_n\}, n \in \mathbb{N}^+\} = \lim_{n \to \infty}\sum_{i = 1}^n a_i > \lim_{n \to \infty}na_1 = \infty$$. This means that $$\sum P = \infty$$ since $$\{\sum F_n: F_n = \{a_1,...,a_n\}, n \in \mathbb{N}^+\} \subset \{\sum F: F\text{ is a finite subset of } P\}$$.

I think my proof is a bit clumsy but is it correct?

Suppose $$P$$ is uncountable, $$\exists \epsilon_1 > 0$$, s.t. $$P_1 = [\inf P + \epsilon_1, \sup P]\cap P$$
This is certainly a true statement! The issue is that you haven't argued why such an $$\epsilon_1$$ exists. And, if you do manage to argue that such an $$\epsilon_1$$ exists, then you are basically done with the proof: just pick a sequence of points $$x_n \in P_1$$, and $$\sum_{k=1}^n x_k \ge n(\epsilon_1 + \inf P) \ge n\epsilon_1 \to \infty$$ as $$n \to \infty$$. This immediately shows that $$\sum P = \infty$$.
Instead, I suggest defining $$P_n = P \cap [1/n, \infty)$$. Argue that $$P_n$$ is finite, and $$P = \bigcup_{n=1}^\infty P_n$$.