the proof of the value of angular frequency must be rational number

Sinusoidal Signal
All continuous-time signals are periodic. The discrete-time sinusoidal sequences may or may not be periodic. They depend on the value of ω. For a discrete time signal to be periodic, the angular frequency ω must be a rational multiple of 2π.

下面是证明：

另外通过欧拉公式：
e i x = cos ⁡ x + i sin ⁡ x e^{ix} = cos x + isin x eix=cosx+isinx,
可解释图中：
e j Ω o N = 1 = > Ω o N = 2 π n e^{jΩoN}=1 => ΩoN=2πn ejΩoN=1=>ΩoN=2πn