I've just started learning these things and so probably my questions will be very easy. Please forgive me.


A metric space $(X,d)$ is called locally finite if every bounded set is finite.
A metric space is said to have coarse bounded geometry if there is $\Gamma\subseteq X$ such that


1) there exists $c>0$ such that the set of points $x\in X$ such that $d(x,\Gamma)\leq c$ is dense in $X$. 


2) For all $r>0$, there exists $K\_r$ such that, for all $x\in X$, $|\Gamma\cap B\_r(x)|\leq K\_r$, where $B\_r(x)$ stands for the ball of radius $r$ about $x$.



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> **Question 1:** what is an example of metric space without coarse bounded geometry? 
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Well, infinite dimensional Banach spaces. But I would like something *more handable*.



> 
> **Question 2:** Is it true that locally finiteness implies coarse bounded geometry?
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Maybe I have misunderstood, but in a published paper I have found a sentence that looks implicitly assume that the answer is positive. It might be trivial, but I am not quite convinced. 


Thanks in advance,


Valerio



Local finiteness and coarse bounded geometry
Q2--no. Let $A\_n$ have cardinality $n+1$ for $n=0,1,...$. Specify all distances between distinct points in the same $A\_n$ to be one, and the distance between a point in $A\_n$ to a point in $A\_m$ to be $n+m$ when $n\not= m$. 


This gives a simple example for Q1 as welll.


