If $e:=\lim_{n\to\infty} (1 + \frac{1}{n})^n$, prove that $$ \lim_{n\to\infty} \left(1-\frac{1}{n}\right)^{-n} = e $$ without using the property that says: if $\lim_{n\to\infty} a_n = \infty$, $\lim_{n\to\infty} x_n = x$, then $\lim{n\to\infty}(1+\frac{x_n}{a_n})^{a_n} = e^x$...
I've tried rewriting $1-\frac{1}{n}$, and operating $\lim_{n\to\infty}(1-\frac{1}{n})^n = 1 / \lim_{n\to\infty}(1+\frac{1}{n})^n $ but I couldn't prove it. Any hint? Thanks in advance.
\begin{gather} \lim\limits_{n\to\infty}\left(1-\dfrac{1}{n}\right)^{-n}=\lim\limits_{n\to\infty}\left(\dfrac{n-1}{n}\right)^{-n}= \\ =\lim\limits_{n\to\infty}\left(\dfrac{n}{n-1}\right)^{n}=\lim\limits_{n\to\infty}\left[\left(1+\dfrac{1}{n-1}\right)^{n-1}\left(1+\dfrac{1}{n-1}\right)\right]=e \end{gather}
Note that $$\left(1-\frac{1}{n}\right)\left(1+\frac{1}{n}\right)=1-\frac{1}{n^2}$$
And $$1\geq \left(1-\frac 1{n^2}\right)^n \geq 1-\frac{n}{n^2}$$
The last is an application of Bernoulli's inequality, with $x=-\frac{1}{n^2}$ and $r=n$.
So $(1-\frac 1{n^2})^n\to 1$ and hence $(1-1/n)^n\to e^{-1}$. and $(1-1/n)^{-n}\to e$
We have $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^{-n}=\lim_{n\to\infty}\left(1+\frac{(-1)}{n}\right)^{n\cdot(-1)}.$$ Now, we can bring the limit inside the $(\cdot)^{-1}$ to have $$\left[\frac{1}{e}\right]^{-1}=e.$$ Observe that I used what you have written so that $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}.$$ For the proof of the last equality, see @amWhy's comment.
To prove that $(1-\frac{1}{x})^{-x} = e$, as x goes to infinity, you would have to prove that $ln(1-\frac{1}{x})^{-x}=1$ under the same conditions.I have taken natural log of both sides.
I will now use y=1/x for convenience. In other words,I'd have to prove that when x=1/y, y going to zero:
$(1/-y)*ln(1-y) =(ln(1-y))/(-y)=1$
We have an indefinite form here.Infinity in the numerator, 0 in the denominator.so we can use L'Hopitals Rule . We take the derivatives of both numerator and denominator:
$(-1/(1-y))/-1)$, and this gives 1/1 when y goes to zero.In other words, when x ,being 1/y, goes to infinity , $ln(1-\frac{1}{x})^{-x}=1$ in the limit, which had to be proven.
