Let $A$ be a bounded Borel subset of $\mathbb{R}$. Suppose I have a sequence of finite collections of Borel subsets of $A$, say $(U_{j}^{n})_{j \in \{1 , ... , n \} , n \in \mathbb{N}}$ satisfying the following:
$\bullet $ for every $n$, $(U_{j}^{n})_{j = 1 , ... , n}$ forms a finite partition of $A$,
$\bullet $ $\max_{j = 1 , ... , n } \operatorname{diam}(U_{j}^{n}) \to 0$ as $n \to \infty $,
where $\operatorname{diam}$ denotes the diameter (the supremum of the distances between points in the set). I have two questions about this:
$1$. Is it true that the sequence $ a_{n} = \sum_{j=1}^{n}\operatorname{diam}( U_{j}^{n} ) $ is bounded?
$2$. On a related question, is it true that for every $\epsilon > 0$ it holds $$ \sum_{j=1}^{n}\operatorname{diam}(U_{j}^{n})^{1+\epsilon} \to 0 ,$$ as $n \to \infty$ ?
The results hold if the sets involved are intervals and subintervals, but I need to know if this could hold for arbitrary Borel (finite) partitions of bounded Borel sets.
Thank you very much.