I want to check if my understanding of limsup and liminf is correct. I know that a limsup and liminf always exist even for divergent sequences and to find them, I need to find a monotonically decreasing and increasing subsequences which converge to limsup and liminf respectively. But what if a sequence $a_n$ is defined differently for different values of $n$?
To be more specific, suppose I have this sequence:
$$ \begin{equation} (a_n)_{n\in\mathbb{N}} = \begin{cases} x_n & n=3k-2\\ y_n & n=3k\\ z_n & n=3k-1 \end{cases} \end{equation} $$ with $k\in\mathbb{N}$ and $x_n, y_n$ be alternating sequences.
To find the limsup and liminf of this sequence, is it appropriate to calculate the limits as $n\rightarrow\infty$ for all the subsequences and just say that the largest and lowest limits are limsup and liminf respectively? I hope that this is not too vague, please let me know if I need to provide a concrete example, I wanted to find a general way of solving these types of problems.