Find the error in the following induction proof
Claim: $k$ arbitrary points in a plane always lie on one line
Base case. for $k=1$ and $k=2$ this is apparently correct
Assumption: Claim already shown for $k$ points
Induction step: Let $k+1$ points, $P_1,...,P_{k+1}$ be given, due to the assumption the points $P_1,...,P_{k}$ lie on the line g and $P_2,...,P_{k+1}$ lie on line h, since $P_2$ and $P_k$ both lie on line g and line h , it follows that $g=h$, thus $P_1,...,P_{k+1}$ are all on one line.
So I am not sure whether my assumption is correct but I think the error lies in the case k=3, which was never checked in the base case? I do not know how to argue here correctly.