Finding sup(S) and inf(S)

Question

Am I doing this right?

1. S={$(-1)^n$|n$\in$natural numbers}
   I got that
   S =\begin{cases} -1, & \text{if $n$ is odd} \\ 1, & \text{if $n$ is even} \end{cases}

2. S={$(-1)^n$n|n$\in$natural numbers}
   I got that
   S =\begin{cases} -\infty, & \text{if $n$ is odd} \\ \infty, & \text{if $n$ is new} \end{cases}

Answer 1

The supremum and infimum of a set are numbers, when they exist; you can tell from this a priori that your answers can't be correct. ($\sup S$ is never a case statement, as you've given -- it's either a number, or else it doesn't exist).

For 1), note that $x \leq 1$ for all $x \in S$. Further, $1 \in S$. These together imply that $\sup S = 1$. The same argument shows that $\inf S = -1.$

For 2), the important point is that for every $N \in \mathbb{N}$, there exists $x \in S$ with $|x| > N$. This implies that $\sup S$ does not exist, and similarly for the infimum.