Proving $\lim_{x \to \infty} \left(1 – \left(1 – \frac{1}{x}\right)^x\right)^x=0$ by using $\lim_{x \to \infty}\left(1-\frac{1}{x}\right)^x=\frac1e$

Question

I'm interested in

$$\lim_{x \to \infty} \left(1 – \left(1 – \frac{1}{x}\right)^x\right)^x \tag1$$

I know this should go to $0$, and the way I want to argue this is that
$$\lim_{x \to \infty} \left(1 – \frac{1}{x}\right)^x = \frac1e \tag2$$
so the entire expression looks like
$$\lim x_{\to \infty} \left(1 – \frac1e\right)^x = 0 \tag3$$
How can I formalize this argument? Or what is another way to find this limit?

Answer 1

For large enough $x$ we have $$0<1-\left(1 - \frac{1}{x}\right)^x <.9$$ now use sandwich lemma.