Prove that if q is a quadratic form $\exists x,y \in V : q(x) \gt 0,\phantom{2} q(y) \lt0 \Rightarrow \exists z \in V : q(z) = 0$

Question

I'm trying to prove that for a vector space $V$ of dimension $n \ge 2$

Let $q : V \rightarrow \mathbb{R}$ a quadratic form

$\exists x,y \in V : q(x) \gt 0,\phantom{2} q(y) \lt0 \Rightarrow \exists z \in V : q(z) = 0$

The proof seems trivial but I'm not sure how to start, any suggestions?

Answer 1

$q$ is continuous implies that $q(V)$ is connected. The connected subspace of $\mathbb{R}$ are intervals. This implies that $[q(y),q(x)]\subset q(V)$.