Simple examples of proper metric spaces? I've encountered the term of a "proper" metric space(a metric space is called proper if every closed, bounded subspace is compact), which struck as quite an interesting one, but I can't find any good examples other than $ \mathbb{R}^n $. I've come across this paper: http://www.math.ku.dk/~haagerup/publications/proper_metric_preamble.pdf
but it seems to require a decent of knowledge of alebraic topology, which I have no clue about.
Are there any fairly elementary examples of such spaces?

So, examples should be simple and good. Two undefined requirements... Take your pick.


*

*Every finite set, with an arbitrary metric. 

*Every closed polygon in the plane: triangle, square.

*Every closed polyhedron in $\mathbb R^3$.

*More generally, every closed subset of $\mathbb R^n$.

*Yet more generally, the image of a closed subset of $\mathbb R^n$ under a bi-Lipschitz map.

*Every graph (possibly infinite) of bounded degree, with path metric (distance between two vertices is the length of shortest path between them). In particular, the Cayley graph of a finitely generated group. 

*Every complete Riemannian manifold: sphere, torus, Klein bottle, projective space... 

*Every complete Finsler manifold.

*Every complete sub-Riemannian or sub-Finsler manifold. 

*A finite product of any of the above. 


Loosely speaking, if you come across a metric space that is 


*

*complete, and 

*does not come from functional analysis


then it's probably proper. 
