Finding the Limit of a Sequence (No L'Hopital's Rule) Okay, I feel almost silly for asking this, but I've been on it for a good hour and a half. I need to find:
$$\lim_{n \to\infty}\left(1-\frac{1}{n^{2}}\right)^{n}$$
But I just can't seem to figure it out. I know its pretty easy using L'Hopital's rule, and I can "see", that it's going to be $1$, but apparently it's possible to compute using only the results I've proved so far. That is, the sandwich theorem, the addition/subtraction/multiplication/division/constant multiple limit laws and the following standard limits:
$\displaystyle\lim_{n \to\infty}\left(n^{\frac{1}{n}}\right)=1$
$\displaystyle\lim_{n \to\infty}\left(c^{\frac{1}{n}}\right)=1$, where $c$ is a real number, and
$\displaystyle\lim_{n \to\infty}\left(1+\frac{a}{n}\right)^{n}=e^{a}$.
As well as those formed from the ratio of terms in the following hierarchy:
$$1<\log_e(n)<n^{p}<a^{n}<b^{n}<n!<n^{n}$$
Where $p>0$ and $1<a<b$
e.g. $\displaystyle\lim_{n \to\infty}\left(\frac{\log_e(n)}{n!}\right)=0$
At first I thought maybe I could express the limit in the form of the $e^{a}$ standard limit, but I couldn't seem to get rid of the square on $n$. I've also tried the sandwich theorem, it's obviously bounded above by $1$, but I just couldn't find any suitable lower bounds. I'd really appreciate some help, or even just a little hint, thanks!

Let's use Bernoulli's Inequality to get $$1 - \frac{1}{n} \leq \left(1 - \frac{1}{n^{2}}\right)^{n} \leq 1\tag{1}$$ and then applying squeeze theorem as $n \to \infty$ we get the desired limit as $1$. Note that this particular limit is purely algebraic one and does not need any significant theorems / results dealing with $e$ or $\log$ function.

Bernoulli's Inequality: If $x \geq -1$ and $n$ is a non-negative integer then $$(1 + x)^{n} \geq 1 + nx\tag{2}$$ Here we have used this inequality with $x = -1/n^{2}$ to get the first inequality in $(1)$. The second inequality in $(1)$ is obvious as $0 \leq (1 - 1/n^{2}) \leq 1$ and hence any positive integral power of $(1 - 1/n^{2})$ does not exceed $1$.
