Prove that $\left(1 - \frac{1}{n}\right)^{n-1} \ge e^{-1}$

Question

$\left(1-\frac1n\right)^{n-1} \ge e^{-1}$

I think it's true that $e^{-1} \ge \left(1 - \frac1n\right)^n$

Is that at all useful?

Thanks!

Answer 1

First, we know $1+x\le e^x\,\,\,\forall x\in \mathbb{R}$. Then, if $n\neq 1$ $$1+\frac{1}{n-1}\le e^\frac{1}{n-1}$$ $$\iff \left(1+\frac{1}{n-1}\right)^{n-1}\le e$$ $$\iff \left(1-\frac{1}{n}\right)^{n-1}\ge e^{-1}$$

Note: I just explicitely used the inequality medicu took for granted in his comment.