VC dimensions: Let ${x_1, \ldots, x_N}$ be $N$ labelled points on $\mathbb{R}$, then there exists a sinusoid that separates these points

Question

(Proposition, pg 26): Let ${x_1, \ldots, x_N}$ be $N$ points on $\mathbb{R}$, $N \in \mathbb{Z}$, labelled either $+1$ or $-1$ , then there exists a function from the set $\{t \mapsto \sin(\omega t)| \omega \in \mathbb{R}\}$ that separates these points into two sets, one with all labels ${+1}$, and one with all labels ${-1}$

Does anyone see how to prove this proposition?

Right now I am using induction, however, I cannot see how it can extend to the $N$ point case.

It is also difficult to picture how this works for the $N$ point case.