To be specific, my question is as follows:
Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold
$\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some compatible metric $d$ on $X$?
To be specific, my question is as follows: Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold $\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some compatible metric $d$ on $X$? 


I really should avoid answering questions late at night. My original answer is muddled enough to not work. But here is what it should have been: Let $X_0 = \{0,1\}$ and $X_i = X_{i1} \times X_0$. Give each $X_i$ the $2$adic ultrametric. That is the distance between two sequences of $0'$s and $1'$s is $2^{n}$ where $n$ is the number of share initial symbols the two sequences share. The projections maps are given by truncation. The inverse limit is now $\mathbb{Z}_2$ which has box counting dimension equal to one. 

