Recursion for the Factorial of a Natural Number

In the setting of The Real Numbers: Standing Notation and Background, let $n!$ be the factorial of $n\in\mathbb{N}$. It is the finite product $\prod_{i=1}^{n}i$, and that definition admits only real factors, so $1,\dots,n$ there, and every other natural number occurring below, are read in $\mathbb{R}$ through the canonical map of The Real Numbers: Standing Notation and Background (i); in particular $n!$ is a real number.

(i) Then

$$1!=1, \qquad (n+1)!=(n+1)\cdot n!\qquad(n\in\mathbb{N}).$$