Let $(d_j)$ be a sequence of metrics on a compact space $X$ such that there exists $c,C>0$ with$$ cd_0(x,y) \leq d_j(x,y) \leq Cd_0(x,y) $$for all $x,y\in X$.Suppose also that $(d_j)$ converge uniformly to a metric $d$.I want to show that, for any $S$ Borel subset, $\delta > 0$ and $\varepsilon > 0$,$$ \mathcal{H}_{d_j,\lambda\delta}^r(S) \leq \mathcal{H}_{d, \delta}^r(S) + \varepsilon$$where $\lambda = C/c$ and$$ \mathcal{H}^r_{d,\delta}(S) = \inf\{ \sum (\operatorname{diam}_d U_i)^r \mid S \subset \cup U_i, \operatorname{diam} U_i \leq \delta \}. $$However when I try I always get a factor $(c/C)^r$ and I don't know how to get ride of it is not clear if we can rescale our cover.$$\begin{aligned}\mathcal{H}^r_{d,\delta}(S) + \varepsilon &\geq\sum (\operatorname{diam}_d U_i)^r \\&\geq \sum \left( \frac{c}{C} \operatorname{diam}_{d_j} U_i\right)^r\end{aligned}$$
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