So we want to prove this theorem from Group Theory:
Let $x$ be an element of a group $(G, \circ)$. If $x$ has finite order $n$, then the $n$ powers
$$e, x, x^2, ..., x^{n-1}$$
are distinct, and these elements repeat indefinitely every $n$ powers in the list of consecutive powers of $x$.
Then my book uses a proof by contradiction to show that the powers are unique. After that they continue to show that the powers repeat every $n$ elements:
Consider any power of $x$ of the form $x^{kn}$. That is, any power where the exponent is an integer multiple of the order $n$ of $x$. We have:
$$x^{kn} = (x^n)^k = e^k = e$$
So in the list of consecutive powers of $x$, the element $e$ is repeated every $n$ elements.
Because each element in the list is obtained from the previous element by composing it with $x$, it follows that all the elements in the list repeat every $n$ elements.
I'm grappling with the sentence in bold. I agree that the proof shows that $e$ repeats every $n$ powers.
I also agree that each element in the list of consecutive powers is obtained from the previous element by composing it with $x$.
But somehow I can't get why this implies that all elements repeat.
(I know it's true, I just find it hard to accept the proof in the book shows it properly)
Can someone add some insight why the proof given is sufficient?
The bold statement is just saying, since $x^{kn} = e$, composing with $x$ on both sides shows how all elements repeat:
$$\begin{align*} x^{kn} &= e\\ x^{kn+1} &= x\\ x^{kn+2} &= x^2\\ &\vdots\\ x^{kn+i} &= x^i \end{align*}$$