<s>[INST] Prove that $a_n=(1-\frac{1}{n})^n$ is monotonically increasing sequence I try to solve it Bernoulli inequality but it too complicated, am I missing something easier?
My try-
$$\frac{a_n}{a_{n+1}}=\frac{(1-\frac{1}{n})^n}{(1-\frac{1}{n+1})^{n+1}}\\=(\frac{1}{1-\frac{1}{n+1}})(\frac{\frac{n-1}{n}}{\frac{n}{n+1}})^n\\=(\frac{1}{1-\frac{1}{n+1}})(1-\frac{1}{n^2})^n<1\\\iff (1-\frac{1}{n^2})^n<1-\frac{1}{n+1}$$
here is where that I want to use Bernoulli...
 [/INST] With Bernoulli, but rewriting to make it simpler:
\begin{align}
&\Bigl(1-\frac1{n+1}\Bigr)^{n+1}>\Bigl(1-\frac1{n}\Bigr)^{n}\iff\Bigl(\frac n{n+1}\Bigr)^{n+1}>\Bigl(\frac{n-1}{n}\Bigr)^{n}\\
\iff&\Bigl(\frac{n^2}{n^2-1}\Bigr)^{n}>\frac{n+1}n=1+\frac1n\iff\Bigl(1+\frac{1}{n^2-1}\Bigr)^{n}>1+\frac1n.
\end{align}
Now, by Bernoulli's inequality,
$$\Bigl(1+\frac{1}{n^2-1}\Bigr)^{n}>1\frac{n}{n^2-1}>1+\frac{n}{n^2}=1+\frac1n.$$
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