Clarification in a proof that $n!\leq n^n$

Question

Can someone explain me,I am not able to understand last line (underlined)

Answer 1

1. $(n+1)!=(n+1)n!$ is valid by definition.

2. $(n+1)n! \le (n+1)n^n$ comes from the induction hypothesis.

3. From $n<n+1$ we get $n^n <(n+1)^n$ and therefore $(n+1)n^n<(n+1)^{n+1}$

Answer 2

By definition $(n+1)!=(n+1)n!$.

By hypothesis, $n!\le n^n$ and so $(n+1)n!\le (n+1)n^n$.

Then, since $n<(n+1)$, $n^n<(n+1)^n$ and so $(n+1)n^n<(n+1)(n+1)^n=(n+1)^{n+1}$