The Factorial: Recursion, Positivity and Bounds by Powers

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let factorials be as in The Factorial of a Natural Number with Zero §factorial and powers in $\mathbb{N}_{0}$ as in Powers with Exponents in the Natural Numbers with Zero §power and Powers with Exponents in the Natural Numbers with Zero §zero. Let $n\in\mathbb{N}_{0}$.

¶ $0!=1$ and $(n+1)!=(n+1)\cdot n!$.

¶ $1\le n!\le(n+1)!$.

¶ $2^{n}\le(n+1)!$.

¶ $(n+1)!\le(n+1)^{n+1}$.