Prove that the sequence (1+1/n)^(n+1) is monotonically decreasing,without using functions and derivatives.
$\displaystyle \begin{align*}\left( {1 - \frac{1}{n}} \right)^n\left( {1 + \frac{1}{n}} \right)^n&= \left( {1 - \frac{1}{n^2}} \right)^n \ge\left( {1 - \frac{1}{n}} \right)\\ \left( {1 + \frac{1}{n}} \right)^n &\ge \left( {1 - \frac{1}{n}} \right)^{1-n}\\\left( {1 + \frac{1}{n}} \right)^n &\ge\left( {1 + \frac{1}{n-1}} \right)^{n-1} \end{align*}$.