Signum Function on Natural Numbers as Characteristic Function

Definition
Let $\operatorname{sgn}: \N \to \N$ be the signum function on the natural numbers.

Let $\chi_{\N_{>0}}: \N \to \left\{{0, 1}\right\}$ be the characteristic function of $\N_{>0}$, where $\N_{>0} = \N \setminus \left\{{0}\right\}$.

Let $n \in \N$.

Then:
 * $\operatorname{sgn} \left({n}\right) = \chi_{\N_{>0}} \left({n}\right)$

Proof
The signum function $\operatorname{sgn}: \N \to \N$ is defined as:
 * $\forall n \in \N: \operatorname{sgn} \left({n}\right) = \begin{cases}

0 & : n = 0 \\ 1 & : n > 0 \end{cases}$

The result follows by definition of the characteristic function.