Školjka

%V0 Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality $$(n + 1)h_n+1 - nh_n > r.$$ Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n \geq 3.$