converge 0-regularly

A sequence of sets {A_n} contained in a metric space X with a metric d is said to converge 0-regularly to its limit $A = \mathrm{Lim}\,A_n$ (see hyperspace ) provided that for each $\varepsilon > 0$ there is a $\delta > 0$ and there is an index $n_0 \in \mathbb{N}$ such that if n > n₀ then for every two points $p, q \in A_n$ with $d(p,q) < \delta$ there is a connected set $C_n \subset A_n$ satisfying conditions $p, q \in C_n$ and $diam_n < \varepsilon$ (see [78, Chapter 9, §3, p. 174]).