Let $A(0,1),B(0,0),C(1,0)$ and $D(1,1)$ be four points in the plane $xOy$. Define $M_3=\{A,B,C\}$ and $M_{n+1}=M_n \cup \left\{Z\epsilon xOy\mid \exists V,W\epsilon M_n\text{ for which }\underset{VZ}{\rightarrow}\,=2. \underset{VW}{\rightarrow}\right\}$, $\forall n\geq 3$ I asked today a similar question, but now I have another one and I don't need the previous question anymore. My question is: how to construct the set $M_n$ using that recurrence, because I need to prove that D in not contained in the set. Thank you!
To understand that recurrence: $M_{n+1}$ is the reunion of $M_n$ and the followin set - the set which contains all points Z in the plane xOy and there exist V,W points from the set $M_n$ such that vector $VZ = 2 \cdot \vec{VW}$.
Every point in every $M_n$ has at least one even coordinate. (Indeed, $A$, $B$, and $C$ have this property, and if $V$ has this property then $Z = V + 2\,\overrightarrow{VW}$ also has this property.) Since $D$ doesn't have this property, it's not in any of the $M_n$.