I imagine that the lub is in $\Bbb{R }$
$\{x \in \Bbb{R} : \sin(x)\geq -1\}$
So since the sin(-1) is not a real number does that mean there isn't a greatest lower bound?
since the set would be unbounded does that mean there isn't a least upper bound?
Best Answer
For all $x \in \mathbb R$ we know $-1 \le \sin x \le 1$ so for all $x\in \mathbb R, \sin x \ge -1$ and $\mathbb R \subset \{x|\sin x \ge -1\} $ and so $ \{x|\sin x \ge -1\} = \mathbb R$.
So you need to find the greatest lower bound and least upper bound of $\mathbb R$.
As $\mathbb R$ is neither bounded above (there is no real number, $k$ so that $k \ge x$ for all $x\in \mathbb R$) or below (there is no real number, $m$, so that $m \le x$ for all $x \in \mathbb R$) there are none.