I'm trying to grasp the idea behind limit superior and limit inferior. Using the concept of subsequential limits, I understand that for a sequence $(x_n)$, the $\limsup x_n$ is the largest limit that a subsequence $(x_{n_k})$ will approach. However, looking at this alternative definition/theorem, I get confused:
Definition:
Suppose $\limsup x_n \in \mathbb R$. Then $\beta = \limsup x_n$ if and only if for all $\epsilon > 0$,
(i) there exists $n_0 \in \mathbb N$ such that $x_n < \beta + \epsilon$ for all $n \ge n_0$ and
(ii) given $n \in \mathbb N$, there exists $k \in \mathbb N$ with $k \ge n$ such that $x_k > \beta - \epsilon$.
Specifically, I am confused as to how this definition works with the sequence $x_n = (-1)^n$.
Obviously, the limit superior of $(-1)^n$ is $1$. So $\beta = 1$. Thus, (i) is satisfied, since $ 1 < 1 + \epsilon$ and $-1 < 1+ \epsilon$. But what about (ii)? It seems that no matter what tail of $x_n$ I consider, I will always end up with $-1$ in $x_k$. And it is not true that $-1>1-\epsilon$ for small enough $\epsilon$.
Can someone explain where my logic fails? Please and thank you.