Let the set $X:=\{(x,y)\in \mathbb{R}^{2}:0\leq x\leq 1,0\leq y\leq 1-x\}$ and for each positive integer $i$ and $j\in\{1,\ldots,2^{i}-1\}$ define the contracttion $f_{ij}:X\longrightarrow X$ by
$f_{ij}:=\Big(\frac{x-j-1}{2^{i}}, \frac{y-2^{i}-j-1}{2^{i}} \Big) $, for each $(x,y)\in X$.
Then, as it is shown in a paper due to N.A. Secelean, the attractor set of the countable iterated function system defined by the functions $f_{ij}$ is a "infinite-type" Sirerpinski triangle. I am trying to estimate the Hausdorff distance from such fractal to the attractor set of the iterated function system defined by a finite number of $f_{ij}$, say $\{f_{11},f_{2,1}, \ldots f_{i2^{i}-1}\}$.
With only he definition of the Hausdorff metric it is not seem clear to state the above estimation so, Somebody know how can we to state an upper bound to the Hausdorff distance between the above sets?
Thank you very much for your time.