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

Question

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

The last line uses the induction hypothesis i.e. $n!\leq n^n$ — Kushal Bhuyan, Mar 14 '17 at 05:16
See also https://math.stackexchange.com/questions/1260233/inductive-proof-that-kkk-for-k-geq-2 — Arnaud D., Nov 29 '19 at 14:12

score 2 · Accepted Answer · answered Mar 14 '17 at 05:18

2

answered Mar 14 '17 at 05:18

Fred

score 2 · Answer 2 · answered Mar 14 '17 at 05:19

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 $(n+1)n^n<(n+1)(n+1)^n=(n+1)^{n+1}$

answered Mar 14 '17 at 05:19

Mark Viola

2 Answers2