Compute using their definition the $\limsup$ and $\liminf$ of $\{x_{n}\}$ with $x_n:= y_1 + y_2 + \dotsc + y_n$
$y_n$ := \begin{cases} 0 & \text{if } n =3p \\ 1 & \text{if } n = 3p+1\\ -1 & \text{if } n= 3p+2 \end{cases} Definitions:
$\limsup{x_n} = \lim{y_n}$ with $\{y_{n}\} := \{\sup{x_m},m \geq n\}$
$\liminf{x_n} = \lim{z_n}$ with $\{z_{n}\} := \{\inf{x_m}, m \geq n\}$
In the same exercise we were asked to compute the $\limsup$ and $\liminf$ of $\{x_{n}\}$ with $x_n:= (-1)^n(2-\frac{1}{n})$. What I did for the $\limsup$ was show that the subsequences $y_{2p}$ and $y_{2p+1}$ such that $y_{2p}= \sup\{2-\frac{1}{2p},-2+\frac{1}{2p+1},\dotsc\}$ and $y_{2p+1}= \sup\{-2+\frac{1}{2p+1},2-\frac{1}{2p+2},\dotsc\}$ both converge to 2 so that $\limsup{x_n} = 2$. Using the same method, I showed that $\liminf{x_n} = -2$.
How do I find the $\limsup$ and $\liminf$ of $\{x_{n}\}$ described above?