Induction principle for $n!How can I prove that $n!<n^n$ for every $n>1$ using the induction principle?
If I put $n=2$ I get $2<4$ so I know that $n!<n^n$ is true.
Now I don't know how to prove that $(n+1)!<(n+1)^{n+1}$. I can write $(n+1)!=n!(n+1)$ but then I don't know how to go on.
 A: HINT
$$(n+1)!=(n+1)\cdot n! \stackrel{\color{red}{n!<n^n}} <(n+1)\cdot n^n\stackrel{\color{red}{n^n<(n+1)^n}} <(n+1)(n+1)^n$$
A: Hint:
$$(n+1)^{n+1} = (n+1)\cdot (n+1)^{n} > (n+1)\cdot n^n$$
A: The key is using the induction assumption $n!\lt n^n$
$$(n+1)!=n!(n+1)\lt (n+1)n^n$$
Now use $n^n\lt (n+1)^n$ to get
$$(n+1)!\lt (n+1)^{n+1}$$
A: Base case:
$$1<n\implies 1!<n^1.$$
Inductive step:
For $k<n$,
$$k!<n^k\implies (k+1)!=(k+1)k!<(k+1)n^n\le n^{k+1}.$$

Then for $k=n$,
$$n!<n^n.$$
A: But, I don't think the problem needs to apply mathematical induction, because the inequality is pretty obvious.
Since $1<n,2<n,3<n,\cdots,n-1<n,n \leq n$ for $n>1$, by multiplying all of these, we have that $$1 \times 2 \times \cdots \times (n-1) \times n <\underbrace{n \times n \times \cdots \times n}_{\mbox{n factors }}.$$
Thus, $$n!<n^n.$$
A: Suppose $n>1$ and that $n!<n^n$ holds. Then you can surely write
$$
(n+1)!=(n+1)n!<(n+1)n^n
$$
You would be done if you can prove that
$$
(n+1)n^n\le(n+1)^{n+1}
$$
which is equivalent to
$$
n^n\le(n+1)^n
$$
which in turn is equivalent to $n\le n+1$, which is obviously true.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
\pars{n + 1}! & = {\pars{n + 1}! \over \pars{n + 1}^{n + 1}}
\,\color{blue}{\pars{n + 1}^{n + 1}} =
\color{blue}{\pars{n + 1}^{n + 1}}\,{n! \over \pars{n + 1}^{n}} =
\color{blue}{\pars{n + 1}^{n + 1}}\,{n\pars{n - 1}\cdots 2 \times 1 \over n^{n}\pars{1 + 1/n}^{n}}
\\[5mm] & =
\color{blue}{\pars{n + 1}^{n + 1}}\,\ \overbrace{\underbrace{{\pars{1 - 1/n}\pars{1 - 2/n}\cdots
\bracks{1 - \pars{n - 2}/n}\bracks{1 - \pars{n - 1}/n} \over \pars{1 + 1/n}^{n}}}_{\ds{{\mbox{numerator}\ < 1} \atop {\mbox{denominator} > 1}}}}^{\ds{\Large\color{red}{< 1}}}
\\[5mm] & <
\bbx{\large \pars{n + 1}^{n + 1}}
\end{align}
