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Why do we use the axiom of choice for finding a non-empty subset of a given set?

I’m having a hard time understanding why the axiom of choice is necessary in the following question,

Question: Is there a non-empty subset $X$ of the set of real numbers with the property that $\forall n \in \mathbb{N}, nx \in X$?

By non-empty I mean that it is $ot\subseteq\emptyset$, and by $n \in \mathbb{N}$ that the $n$ is a natural number.
I’m thinking that if we can have a one-to-one correspondence between $X$ and the natural numbers, then $\mathbb{N} \to X$ would be injective and hence we’d have a proof without the axiom of choice.
My question is: what is the precise reason why we use the axiom of choice for this question?

A:

The axiom of choice is needed because you want to choose an element from every set that you look into (what is usually denoted by the symbol $\bigcup_{n\in\Bbb N}A_n$), not just every element of every set.
In your proposed line of argumentation, the point is that the natural number $0$ corresponds to the empty subset of $\Bbb R$, but the point is that there are larger and larger subsets with the property that $n\cdot x\in X$. This is not possible in a naive fashion, since there is no largest non-empty subset of $\Bbb R$ with this property.
That is, the property is a very strong property that actually requires a different proof than for every $n\in\Bbb N$ there is an $x\in X$ with $n\cdot x\in X$ (which in turn cannot be shown without the axiom of choice).
In fact, you can already see this in the statement of the problem: the statement only looks for a non-empty
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