Let $S=\{ (x,y,z) \in \mathbb{R}^3 | z^2-x^2-y^2-1>0 \}$. Proof that $S$ is an open set.
I'm trying to prove it as follows:
By definition, I need to proof that $\forall u \in S, \exists B_{r} (u) \subset S$ for some $r>0$.
Let $u=(a,b,c) \in S. \iff c^2-a^2-b^2-1>0.$
Then, I tried to find a radius $r>0$ such that there's an open ball centered in $u$ with radius $r$ in many ways (for example, finding the minimun value of $||u-v||$ where $v \in \partial S$), but I was unsuccesfully.
Can someone help me with this?