Use induction to prove that $n! \leq n^{n-1}$ Use induction to prove that $n! \leq n^{n-1}$ for all integers $n\geq 1$. I'm having a hard time with induction and my professor said this is a good future test like question if someone can post a solution and explain it would help me out a lot. Thank you.

Induction or not,
$$n! = 1\cdot 2\cdot 3\cdot\ldots\cdot n \leq 1\cdot n\cdot n\cdot\ldots\cdot n = n^{n-1}.$$
