Why this set is countable?

Question

Why is the following set countable? (someone told me it's apparent but I cannot figure out a rigorous proof)

$$S=\{\sum_{i=1}^nc_ie_i|n\in N^+, c_i\in Q\}$$

where $N^+$ is the set of positive integers, $Q$ is the set of rational numbers and $\{e_i\}$ is a Shauder's basis of a Banach space.

Answer 1

It is more or less explicitly a countable union of countable sets.

It is the union over $n \in \mathbb{N}$ of the same kind of sums but with just exactly $n$ terms.

And the sum with just $n$ terms is a rational linear combination of the first $n$ terms of the basis and so clearly bijects with $\mathbb{Q}^n$, which is countable.

Answer 2

Hint: There is a natural injection that you can make from $S$ to $\mathbb{N} \times \mathbb{Q}$ thus, $S$ is countable if $\mathbb{N} \times \mathbb{Q}$ is countable.