You might begin by breaking the sequence down into four subsequence:
$\begin{align*} a_{4k}&=1+\frac1{4k}\tag{1}\\ a_{4k+1}&=2-\frac1{4k+1}\tag{2}\\ a_{4k+2}&=-1+\frac1{4k+2}\tag{3}\\ a_{4k+3}&=-2-\frac1{4k+3}\tag{4} \end{align*}$
It’s very clear how each of these is behaving:
- $(1)$ is monotonically decreasing from a maximum of $\frac54$ to a limit of $1$;
- $(2)$ is monotonically increasing from a minimum of $1$ to a limit of $2$;
- $(3)$ is monotonically decreasing from a maximum of $-\frac12$ to a limit of $-1$; and
- $(4)$ is monotonically increasing from a minimum of $-\frac73$ to a limit of $-2$.
From this information alone you can already see that the sequence attains its minimum at $a_3=-\frac73$, which must therefore by its infimum. You can also see that $a_n<2$ for all $n$, while $\lim\limits_{k\to\infty}a_{4k+1}=2$, so $2$ must be the supremum of the sequence.
How you proceed from there depends on what you know about limits superior and inferior. It’s clear from $(1)-(4)$ that if a subsequence of $\langle a_n:n\in\Bbb N\rangle$ converges at all, it must converge to $1,2,-1$, or $-2$.
If the subsequence has infinitely many terms in two or more of the subsequences $(1)-(4)$, it does not converge. If it has infinitely many terms in only one of them, it converges to the same limit as that subsequence.
If you know that $\liminf_na_n$ is the smallest $x$ such that $\langle a_n:n\in\Bbb N\rangle$ has a subsequence converging to $x$, then you can immediately conclude that $\liminf_na_n=-2$, the smallest of the limits $-2,-1,1$, and $1$. Similarly, if you know that $\limsup_na_n$ is the largest $x$ such that $\langle a_n:n\in\Bbb N\rangle$ has a subsequence converging to $x$, then you can immediately conclude that $\liminf_na_n=2$, the largest of the four subsequential limits.
If you have to work directly from the definitions of $\liminf$ and $\limsup$, you can’t do much better than what you actually did, though you might find it easier to state in terms of the four-way split into subsequences.