I have an infinite number of sets $R_1, R_2, R_3, \dots $ where $R_1 \subset R_2 \subset R_3\subset \dots$
Each set is the output (image / range) of a corresponding function:
$$R_i = Image(T_i)$$
$F_i$ has i inputs $(x_1,\dots, x_i)$ with all inputs $x_i \in \mathbb{Z}_{>0}$
For a lack of a better term, the recurrence relation for the functions interpolates. Examples may be easier to understand.
$T_1(x_1) = \frac{1}{3}(2^{x_1}-1)$
$T_2(x_1,x_2) = \frac{1}{3}(2^{x_1}\times \frac{1}{3}(2^{x_2}-1)-1)$
$T_3(x_1,x_2,x_3) = \frac{1}{3}(2^{x_1}\times \frac{1}{3}(2^{x_2}\times \frac{1}{3}(2^{x_3}-1)-1)-1)$
Each function is a subset of the next with the last input being 2.
$T_n(x_1, \dots, x_n) = T_{n+1}(x_1,\dots,x_n,2)$
I'm after some notation that will help me probe these sets as I'm trying to find the output (range / image) of the infinite-th T $(T_\infty)$ and ideally show that $T_\infty \supset \mathbb{O}$ where $\mathbb{O}$ is the set of positive odd numbers.