$X \subset \mathbb{R}_{>0}$ exists $C$ for every $\{x_1, ..., x_n\} \subset X$ $\sum^{n}_{i=1} x_i<C$ Then $X$ is countable

Question

Let $X \subset \mathbb{R}_{>0}$ such that exists $C$ that, for every $\{x_1, ..., x_n\} \subset X$ $$\sum^{n}_{i=1} x_i<C.$$ Prove that $X$ is countable

My intuition goes that if $X$ is not countable, there must exist some $c>0$ such that there are infinite elements of $X$ above $c$. If this is the case, then it's easy, since exists a natural number $n$ such that $n\cdot c > C$, and I have more than $n$ elements.

The best thing I'm thinking, and I'm not sure if it's right is something like this:

There are finite elements of $X$ that are greater than $1$. If not, I can take $[C+1]$ of these elements and the sum is greater than $C$.

With the same reasoning, I can do the same with $1/2$, $1/4$,...

I'm not sure if this ends up saying that X should be countable.

Answer 1

Hint: define $X_k=X \cap (1/k,\infty)$ and note that $X=\bigcup_{k=1}^\infty X_k$. If you can prove that all the $X_k$ must be at most countable then you're done. (This is basically following your intuition but in a more systematic way.)