Convergent Trigonometric Series is Periodic

Theorem
Let $S \left({x}\right)$ be a trigonometric series:


 * $S \left({x}\right) = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \left({a_n \cos n x + b_n \sin n x}\right)$

Let $S$ be convergent.

Then $S$ is periodic:


 * $\forall r \in \Z: S \left({x + 2 r \pi}\right) = S \left({x}\right)$

Proof
Let $S \left({x}\right)$ converge to some $L \in \R$.

Let $r \in \Z$ be arbitrary.

Then: