The members of the sequence can be seen as follows: $(-1)^n - \dfrac{1}{n}= \Bigg\\{\begin{matrix}\big(1-\dfrac{1}{n}\big),& \text{when n is odd}\\\ \big(-1-\dfrac{1}{n}\big),&\text{when n is even} \end{matrix}$
When $n$ is odd, the value of $\big(1-\dfrac{1}{n}\big)$ increases as $n$ increases and thus the supremum (for odd $n$) is obtained as $n \to \infty$, and the limit is $lim_{n\to \infty}\big(1-\dfrac{1}{n}\big) = 1$. The infimum (for odd $n$) is obtained for $n = 1$, that is, $1 - 1 = 0$.
Similarly, when $n$ is even, the value of $\big(-1-\dfrac{1}{n}\big)$ increases as $n$ increases and supremum (for even $n$) is obtained as $n\to \infty$, with the limit being $lim_{n\to \infty}\big(-1-\dfrac{1}{n}\big) = -1$ and infimum (for even $n$) being $-2$ obtained for $n = 1$.
So, for the given sequence $\big\\{(-1)^n - \dfrac{1}{n}\big\\}$, the infimum is $-2$ and the supremum is $1$.