Show that length of longest edge in Delaunay triangulation goes to zero

Question

Suppose that we have a set of vertices $V$ in a certain space $\Omega$ and that we iteratively add a vertex to $V$ and build a Delaunay triangulation on these vertices. Now, if $\mathcal{T}^1$, $\mathcal{T}^2$, ... denotes the corresponding sequence of Delaunay triangulations, I would like to show that the length of the longest edges in these triangulations, denoted by $|\mathcal{T}^j|$ goes to $0$ as $j \rightarrow \infty$. I am wondering if this is even true and what conditions the space $\Omega$ needs to satisfy to obtain this result. Thanks in advance for your help!