Let $\Omega$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,x_i\neq y_i\}$. Now define a function $\pi:\Omega\rightarrow [0,1]$, where $\pi((x_n))$ is the number in $[0,1]$ that has $n$th digit $x_n$ in its binary expansion.

Does anyone have a reference for published work in which it is proven that the Hausdorff dimension of a set $X$ in the metric space $(\Omega,d)$ is equal to the Hausdorff dimension of $\pi(X)$ in $[0,1]$ with the Euclidean metric?