Given is the following sequence $a_n=(-n)^{(-n)^n}$ $(n \in \mathbb{N})$. Find all limit points.
Here's what I have so far, I divided it in three cases.
Case 1: $-n > 0 \rightarrow n < 0$
I did this "trick" $m=-n, a_n=m^{m^{-m}}$
$m^{-m}$ converges to $0$ and therefore, $a = m^0 = 1$, when $n$ tends to infinity. This is our only limit point.
Case 2: $-n < 0 \rightarrow n > 0$
I divided into two more cases here:
n is even: $-n^n$ diverges to negative infinity, therefore $a_n=(-n)^{(-n)^n}$ also. n is odd: $-n^n$ diverges to positive infinity, therefore $a_n=(-n)^{(-n)^n}$ also.
Is this correct? Thanks