Consider a convex polytope $P$ in $\mathbb{R}^{d}$ as the convex hull of a minimal finite set $V$ whose elements are called "vertices", or equivalently as the bounded intersection of a finite collection of half-spaces.
A half-space is defined as $H^{\color{red}{\le}}:=\{x\in\mathbb{R}^{d}\mid c^{T}x\color{red}{\le} \delta,\, \delta\in\mathbb{R},\,c\in\mathbb{R}^{d}\setminus\{0\}\}$. Any hyperplane $H:=\{x\in\mathbb{R}^{d}\mid c^{T}x= \delta,\, \delta\in\mathbb{R},\,c\in\mathbb{R}^{d}\setminus\{0\}\}$ defines two half-spaces. A half-space containing $P$ is said valid for $P$.
A face $F$ of $P$ is the intersection between a valid half-space and $P$, which gives a set of the form $\{x\in P\mid c^{T}x= \delta,\, \delta\in\mathbb{R},\,c\in\mathbb{R}^{d}\setminus\{0\}\}$ or $P$ itself. The dimension of a face $F$ is the dimension of the affine hull of $F$.
With the above informations only (no other result on polytopes), we want to prove that any face $F\neq P$ is contained in some face $G$ with $\text{dim } G=\text{dim } F+1$.
The problem is I cannot understand how to begin. Even if I admit that a vertex is a face, I do not get how I can prove the existence of any proper face... I don't ask for a full resolution, only for hints on how to begin with such a problem. Thank you.