Zkusíme, o co jde :
  {
   \begin{eqnarray*}
      {(n+1)}^n                     & < &n^n.n  \\
      {\left(\frac{n+1}n\right)}^n  & < &n      \\
      {\left(1+\frac1n\right)}^n    & < &n      \\
      {\left(1+\frac1n\right)}^n < e& < &3\le n \\
      &\Downarrow&\\
      {(n+1)}^n                     &\le&n^{n+1},\quad {\rm pro\ } n\ge3.\ \checkmark
   \end{eqnarray*}