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 the period 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 $X = \mathbb C$

The case for when $n=0$ is trivial, because
 * $f \left({x + 0 \cdot L}\right) = f \left({x}\right)$

For Positive Integers
For $n \in \mathbb Z_{\gt 0}$, the result will be proved by induction.

Basis for the Induction
The case for which $n=1$ is already given by the definition of a periodic function.

This will be our basis for the induction.

Induction Hypothesis
For some $n \in \mathbb Z_{\gt 0}$, suppose
 * $f \left({x}\right) = f \left({x+nL}\right)$

This will be our induction hypothesis.

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

The result follows from the Principle of Mathematical Induction, because $\mathbb Z_{\gt 0} = \mathbb N_{\gt 0}$.

For Negative Integers
For some $n \in \mathbb Z_{\lt 0}$, it is possible to do:

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

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