Examples of locally finite set whose convex hull is the whole $\mathbb{R}^n$.

Question

In our exercise a locally finite set A is a set such that $A \cap B(r)$ is a finite set for all $r \geq 0$, where $B(r)$ is the ball centered at $0$ and has radius $r$. Shouldn't conv($A$) then be bounded?

Answer 1

Consider $A=\mathbb{Z}^n$. Then $$\vert A \cap B(r)\vert \leq (2r)^n <\infty.$$ But surely the convex hull of $A$ is all of $\mathbb{R}^n$.