I am looking for an example of a separable, locally-compact metric space which is not $\sigma$-compact.
At first I thought I could show that if a metric space is separable and locally-compact, then it must be $\sigma$-compact. But I could not show it and I haven't found any theorem that implies that. So I figured there must be an example of a space which is metrizable, separable, locally-compact but not $\sigma$-compact. Clearly such a space cannot be compact, so I am looking for a locally-compact non-compact metric space.
Can someone give me such an example?