$$
\begin{align}
\lim_{n\to\infty}\left(\frac{e-1}{e}\sum_{k=0}^n\left(\frac kn\right)^n\right)^n
&=\lim_{n\to\infty}\left(1-\frac{e-1}{e}\sum_{k=0}^n\left(e^{-k}-\left(1-\frac kn\right)^n\right)\right)^n\tag{1}\\
&=\lim_{n\to\infty}\left(1-\frac{e-1}{e}\sum_{k=0}^n\left(e^{-k}-e^{-k-\frac{k^2}{2n}+O\left(\frac{k^3}{n^2}\right)}\right)\right)^n\tag{2}\\
&=\lim_{n\to\infty}\left(1-\frac{e-1}{e}\sum_{k=0}^\infty e^{-k}\left(\frac{k^2}{2n}+O\left(\frac{k^3}{n^2}\right)\right)\right)^n\tag{3}\\
&=\lim_{n\to\infty}\left(1-\frac1{2n}\frac{e-1}{e}\frac{e(e+1)}{(e-1)^3}+O\left(\frac1{n^2}\right)\ri…

if there is a statement P(n) where n is a set of natural numbers and I decide to prove P(n) is doing the following sufficient to prove its correctness?

1) Suppose P(n) is not true. Then, there has to be a set C that has values of n that refutes P(n)

2) By WOP set C has to have a minimum for which P(n) is false

3)Assume m is the minimum. And then if I prove that (not)P(m)->(not)P(m-a) then I contradict that m is the minimum.

Hence I prove that the statement that C is empty and hence P(n) is true