Convergent Trigonometric Series is Periodic

Theorem
Let $\map S x$ be a trigonometric series:


 * $\map S x = \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$

Let $S$ be convergent.

Then $S$ is periodic:


 * $\forall r \in \Z: \map S {x + 2 r \pi} = \map S x$

Proof
Let $\map S x$ converge to some $L \in \R$.

Let $r \in \Z$ be arbitrary.

Then: