# Can i get every face of a polytope by taking a facet (of a facet (of a facet (…))) of the polytope?

Let $P$ be a polytope, i.e. a convex subset of a finite-dimensional real vector space with finitely many extreme points. Let $F$ be a proper face of the polytope, i.e. a subset $F \subset P$ such that there exists a closed half-space containing $P$ such that $F = \partial H \cap P$, where $\partial H$ is the hyperplane given by the boundary of $H$.

I wonder if there must be a sequence $P = F_1 \supset F_2 \supset \ldots \supset F_k = F$ of subsets of $P$ such that $F_{i+1}$ is a facet of $F_i$ for every $i \in \{1, \ldots, k-1\}$ (with the definition that a facet of a polytope is a $(d-1)$-dimensional face of a $d$-dimensional polytope).

How can I prove the existence of such a sequence?

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Notice that it is enough —by induction— to show that every proper face is a face of a facet. Can you de that? – Mariano Suárez-Alvarez Nov 27 '11 at 4:55
I agree that this is enough. However, I don't see how to prove this. Can you give me another hint? – Tom Jonathan Nov 27 '11 at 7:56