I'm really not sure where to start. Induction can really be used, and that seems like the only way to prove for all $n$.
I will construct an explicit series of polygons satisfying the conditions. No induction needed.
- Three consecutive vertices of the polygon $A,B,C$ define an equilateral triangle.
- The remaining $n-3$ vertices lie along the arc between $A$ and $C$ with centre $B$ and are joined in sequence.
$\angle B$ is 60° by definition, while $\angle A$ and $\angle C$ are acute because they are the angles between a chord and a radius. For each other vertex, its neighbours form a chord that lies inside the $AC$ chord, so its angle must be greater than 120° and thus obtuse. All angles are less than 180°, so the polygon is convex and the proof is finished.