Can someone help me prove that the sequence
$$\left(1+\frac{1}{N} \right)^{N+1}$$
is decreasing for $N \in \Bbb N $?
EDIT: I have tried the following..
\begin{align} \frac{\left(1+\frac{1}{K}\right)^{K+1}}{\left(1+\frac{1}{K+1}\right)^{K+1}} & =\left(1+\frac{1}{K^{2}+2K}\right)^{K+1} \\ \tag{Bernoulli} & >1+\frac{K+1}{K^{2}+2K} \\ & >1+\frac{K+1}{K^{2}+2K+1} \\ & =1+\frac{1}{K+1} \\ & =\frac{K+2}{K+1} \\ \end{align}
$$\left(1+\frac{1}{K+1}\right)^{K+2}=\frac{K+2}{K+1}\left(1+\frac{1}{K+1}\right)^{K+1}<\left(1+\frac{1}{K+1}\right)^{K+1}$$
