Given two functions $h(x)$ and $f(x)$, with $x \in X \subset \mathbb{R}^6$, with the first $3$ elements of $X$ being a unit vector. I want to prove that a map $F$ given as
$F = \begin{bmatrix} h(x) \\ h(f(x)) \\ h(f(f(x)))\\ \vdots \end{bmatrix}$,
with up to $N$ compositions is injective.
The function $h$ is defined as $h(x) = Cx$, with $C = [I_{3\times3} \, 0_{3\times3}]$.
As for $f$, we have $f(x): X \rightarrow X$, and $f(x) \neq x$.
Applying the definition of infectivity I can prove that for up to $N = 2$, $F$ is invective, i.e.,
$F_1 = \begin{bmatrix} h(x) \\ h(f(x)) \end{bmatrix}$, and $F_2 = \begin{bmatrix} h(x) \\ h(f(x)) \\ h(f(f(x))) \end{bmatrix}$ are injective.
Is there any way I can generalize the infectivity to N compositions?
My idea so far is to try to find an injective function that takes the map for $N = 1$, and gives the map for $N = 2$, and then try to prove for $N$ by induction, but so far I haven't made any progress