I have the following math problem that I got stuck on: Proof that the following inequalities hold for all $n \in \mathbb{N}:$ $$n^{n} * \exp(-n + 1) \leq n! \leq n^{n+1} * \exp(-n + 1)$$
My idea was to use induction, but I have trouble finding the step from $n$ to $n+1$. $$(n+1)^{n+1} * \exp(-(n+1) + 1) \leq (n+1)!$$ Can be rewritten as: $$(n+1)^{n} * (n+1) * \exp(-n) \leq n! * (n+1)$$ Which can be simplified (or atleast I hope it can) to: $$(n+1)^{n} * \exp(-n) \leq n!$$ But then I have no idea how to proceed, let alone do the second inequality.
