I'm stuck in a property when I was reading proof of a theorem about polyhedral theory:
Every full dimensional, bounded and non-empty polyhedron in $\mathbb R^n$ has at least $n+1$ vertices.
A vertex is a point of a polyhedron such that we cannot write it as $\lambda x+(1-\lambda)y$ such that $\lambda \in (0,1)$.
Can anyone give me a hint to prove this property? I think I should consider the fact that this polyhedron is convex hull its vertices and with induction I should prove that convex hull of every $k$ elements is at most a $k-1$ dimensional space. But it doesn't make sense to me also these properties. Could you please help me to understand these fact truly?