Why reflection and rotation are sufficient operations in dihedral group?

Question

I know a bit of elementary group theory but please ignore dihedral group in the title and let's make it simple enough so a high student can read the question and the answer(s)...

Suppose we have a regular polygon with n vertices/sides.

Question: Show that by only rotation(s) and reflection(s) you can turn a given arrangement of numbers on the vertices to any other arrangement.

An incorrect answer: Because there is no other action that we can impose, except for reflection and rotation.

Is this answer a complete and rigorous proof for the question? How to prove the mentioned statement?

Edit - In gp-th language, prove that there always a function of the form $r^{n_1}s^{n_2} \dots r^{n_k}s^{n_k}$ such that any regular numbered polygon can be turned to another desired regular numbered polygon.

Answer 1

Number the vertices $1$ through $n$, with $1$ at the top. When you turn you given arrangement into another arrangement, the vertices must be in the same order or the reverse order (a reflection), and the label at the top may be changed (a rotation). Therefore reflections and rotations are enough to give you all the possible symmetries of a polygon.