TITLE: Limit of $\left(1-\frac{1}{n^2}\right)^n$
QUESTION [8 upvotes]: I'm trying to find the limit of $a_n = \left(1-\frac{1}{n^2}\right)^n$ for $n \rightarrow \infty$.
It seems that the limit is $1$, since $a_n = 0.999...$ for large $n$. The presentation $a_n = \frac{(n^2-1)^n}{n^{2n}}$ and expanding was my first idea, but I couldn't get the result from there. Any ideas?

REPLY [10 votes]: By Bernoulli's inequality
$$ 1 - \frac{1}{n} \leq \left(1 - \frac{1}{n^2}\right)^n \leq 1.$$