In Discrete-Time Signal Processing, Oppenheim writes
For the discrete-time sinusoidal signal $x[n]=A \cos(\omega_0 n + \phi)$, as $\omega_0$ increases from $\omega_0 = 0$ toward $\omega_0 = \pi$, $x[n]$ oscillates progressively more rapidly. However, as $\omega_0$ increases from $\omega_0 = \pi$ to $\omega_0 = 2\pi$ , the oscillations become slower.
Suppose I have the discrete-time signal $$X_n = A\cos(\omega_0 n + \phi).$$
If $N$ is its fundamental period, we have that $\omega_0 N = 2\pi k$ for some integer $k$, and so $k$ must be the smallest integer for which $\frac{2 \pi}{\omega_0}$ is itself an integer. If I let $\omega_0 = r \pi$, where $0 < r < 2$ is rational, how can I show that the period decreases as $r \to 1$? To simplify, I can let $r = 1 - 1/m$, and we have that $$N = \frac{2\pi}{r\pi}k = \frac{2m}{m - 1}k = \left(1 + \frac{m+1}{m-1}\right)k$$ which I believe implies that $k = \frac{m-1}{\text{gcd}(m-1,m+1)}$. I would need to show that this decreases in $m$.
Is there a good geometric picture for why the above fact is true?