I am trying to solve Exercise 9.1 of Munkres
Exercise 9.1. Define an injective map $f:\mathbb{Z}_{+} \rightarrow X^{w}$, where $X = \{0,1\}$ and $X^{w} = \prod_{i=1}^{\infty} X$, without using the choice axiom.
Axiom of choice. Given a collection $\mathcal{A}$ of disjoint nonempty sets, there exists a set $C$ consisting of exactly one element from each element of $\mathcal{A}$; that is, a set $C$ such that $C$ is contained in the union of the elements of $\mathcal{A}$, and for each $A \in \mathcal{A}$, the set $C \cap A$ contains a single element.
My attempt in part of choice axiom follows as
$ f(n) = (x_{1},x_{2},x_{3} ,...) $
where $x_{n} = 1$ and others $x_{i} = 0$.
In reference to axiom of choice, I define
$\mathcal{A} = \{(x_{1},x_{2},x_{3},...) \mid x_{j} \in X \} $
So each element of $\mathcal{A}$ is a $w$-tupla. There is no a set $C$ consisting of exactly one element from each element of $\mathcal{A}$ because I take only those whose nth component is $1$ and others are zero.
Am I right? How can I define this function?