Proposition 1   Suppose $p\colon V\to \sR$ is a seminorm on a real (or complex) vector space $V$ . Then

1. $p(0)=0$ ,
2. $p(v)\ge 0$ for all $v\in V$ .

Proof. Property $1$ follows using homogeneity; $$ p(0)=p(0\cdot 0) = |0| p(0) =0. $$ Property $2$ follows using sublinearity and Property 1; $$ 0=p(0)=p(v-v) \le p(v)+p(-v) = 2p(v). $$