A polytope is the convex hull of a finite set of points in $\mathbb R^n$. The $d$-skeleton of a polytope is the set consisting of faces of dimension at most $d$. I would like to show that every $d$-skeleton of every polytope is strongly connected, meaning that any two $d$-dimensional faces can be connected by a sequence of $d$ dimensional faces, where every two consecutive faces intersect in a dimension $d-1$ face.
For $d=1$, this is just saying that the graph of edges and vertices is connected, which seems relatively easy to prove. But what about larger $d$?