Show that points in plane lie within the interior or on the boundary of a triangle with area less than $4$

I think you're confusing an implication with its converse. The problem asks you to prove an implication concerning a finite set $S$ of points in the plane. The hypothesis in this implication is that, for any three points $A,B,C$ in $S$, the triangle $\triangle ABC$ has area less than $1$. I'll call this hypothesis $H$ for short. The conclusion you're supposed to deduce from this hypothesis is that the whole set $S$ lies within a triangle of area less than $4$. I'll call this conclusion $C$ for short. So you're supposed to prove the implication $H\implies C$.

You've shown (by taking $A,B,C$ very near the corners of a triangle of area $4$) that it's entirely possible for $C$ to be true while $H$ is false. That refutes the implication $C\implies H$. But so what; that's not the implication you were asked to prove.