Why if one have an $\varepsilon$-expansive homeomorphism $T:X \rightarrow X$ ($X$ a compact metric space) and a given partition $D$ of $X$ which has diameter smaller than $\varepsilon$ the sequence of refined partitions $D_n = \bigvee_{i = -n}^n T^{-i} D$ has diameter converging to zero ?

Recall that a $\varepsilon$-expansive homeomorphism $T$ is such that given any two distinct points $x$ and $y$ there exist $n \in \mathbf{Z}$ such that $d(T^nx, T^ny) > \varepsilon$

I can see intituively why this is true, somehow the refined partitions have less an less points in its members precisely because they have diameter less than epsilon but $T$ keeps separating points (and i fact open sets of points) at distance greater than $\varepsilon$.

Thanks in advance!