Injection from the Product of Discrete Sets onto the Unit Interval

Question

Is it possible to construct a injective mapping $$S:=\{(x_1, ..., x_n) | x_i \in \mathcal{N} \} \longrightarrow [0,1]$$ where $\mathcal{N} \subset \mathbb{N}$ is of finite cardinality and $n\in \mathbb{N}$? If so, how would one go about building such a mapping?

Answer 1

Of course, $S$ has also finite cardinality of $m=k^n$ elements if $|N|=k$.

Say $S=\{a_1,...a_m\}$, then map $a_i\mapsto {1\over i}$ is such a map.