Infinite periods in bifurcation diagram

Question

A slightly more theoretical question for you all. Recently I was looking at the logistic map and the resulting bifurcation diagram (shown). Wikipedia says that prior to roughly r = 3.56995, there is a period-doubling cascade that goes from 2 to 4 to 8 to 16, etc. I assume this continues on for infinity. My question is, is there a concrete way to prove this? Ie., prove that there is an infinite amount of even periods prior to roughly r = 3.56995. I was planning on using an induction proof, start proving that the logistic map stabilizes for values 1 < r < 3 (I can attach the completed proof for this case if needed). However, I haven't quite been able to extend this enough to make it complete in induction. Any ideas?