Proving periodic points of tent-map are dense in [0,1]

Question

Let $T$ be the tent map defined by $$T = \begin{cases} 2x \quad \text{ if } x \le 1/2 \\ 2(1-x) \quad \text{ if } x \ge 1/2\end{cases}$$ I am trying to prove that the periodic points of $T$ are dense in the unit interval, $I = [0,1]$. I know that all asymptotically periodic orbits of $T$ are eventually periodic, given that all fixed points are sources. However, I'm struggling in going about solving this given the tools I have. For context, this problem is problem 3.7 in Alligood's text. I was thinking that showing there are infinitely many periodic orbits is sufficient, but I'm unsure. I'm trying to find a more rigorous proof but am at a loss.