TITLE: Prove $n! \leq n^n$
QUESTION [5 upvotes]: Prove by least counter example for all positive integers n. $$ n! \leq n^n$$ 
I keep getting stuck after proving the least element of the set of counterexamples can not equal 1. Any suggestions would be helpful.

REPLY [2 votes]: $n!$ is the number of permutations of $S=\{1, \dots, n \}$, while $n^n$ is the number of functions from $S$ to itself. Permutations are functions, so clearly $n! \leq  n^n$.