General Periodicity Property

Theorem
Let $f: X \to X$ be a periodic function, where $X$ is either the set of real numbers $\R$ or the set of complex numbers $\C$.

Let $L$ be a periodic element of $f$.

Then:
 * $\forall n \in \Z: \forall x \in X: f \left({x}\right) = f \left({x + n L}\right)$

That is, after every distance $L$, the function $f$ repeats itself.

Proof
Let $X = \mathbb C$.

There are two cases to consider: either $n$ is not negative, or it is negative.

Since the Natural Numbers are Non-Negative Integers, the case where $n \ge 0$ will be proved using induction.

Basis for the Induction
The case for which $n = 0$ is trivial, because:
 * $x + 0 \cdot L = x$

Induction Hypothesis
For some $n \in \Z_{\ge 0}$, suppose that:
 * $f \left({x}\right) = f \left({x + n L}\right)$

Induction Step
For the induction step, let $n \to n + 1$.

Then:

Case 2
If $n \lt 0$, then:

Combining the results above, it is seen that for all $n \in \Z$:
 * $f \left({x}\right) = f \left({x + n L}\right)$

The proof for when $X = \R$ is nearly identical to the above proof.