Example of a locally compact metric space which is $\sigma$-compact but not proper Let $(X,d)$ be a locally compact metric space. Then it is known that $X$ is separable if and only if it is $\sigma$-compact (i.e. it can be written as a countable union of compact sets). Moreover, under this condition there is a proper function $f:X\to\mathbb{R}$ and
$$
d'(x,y)=d(x,y)+|f(x)-f(y)|
$$
is a compatible metric (i.e. it generates the same topology) sucht that $(X,d')$ is proper (i.e. every closed ball is compact).
Now here is my question: Is there an example of  a locally compact metric space which is $\sigma$-compact but not proper?

Here is an example with one more property: $X$ is complete. Namely, let $X$ be the real line with the metric $d(x,y)=\min(|x-y|,1)$. At small scales this is the same as Euclidean metric, so the space is complete and locally compact. It is also $\sigma$-compact, as the union of intervals $[n,n+1]$. 
On the other hand, it is not proper since any closed ball of radius $1$ is the entire space, which contains the infinite subset $\mathbb{Z}$ without a limit point.
