Let $\alpha$ be an irrational number with $0<\alpha<1$, and draw a circle in the plane whose circumference has length 1. Given any integer $n \geq 3$, define a sequence of points $P_{1}, P_{2}, \ldots, P_{n}$ as follows. First select any point $P_{1}$ on the circle, and for $2 \leq k \leq n$ define $P_{k}$ as the point on the circle for which the length of $\operatorname{arc} P_{k-1} P_{k}$ is $\alpha$, when travelling counterclockwise around the circle from $P_{k-1}$ to $P_{k}$. Suppose that $P_{a}$ and $P_{b}$ are the nearest adjacent points on either side of $P_{n}$. Prove that $a+b \leq n$.