I am trying to understand the proof of theorem 2.1 from the following paper: https://aif.centre-mersenne.org/item/10.5802/aif.2306.pdf.
Basically the authors prove that given $\alpha=[0;a_1,a_2,\dots,a_l,\dots]$, if the continued fraction starts with arbitrarily long palindromes, then $\alpha$ is transcendental.
The proof is very easy to follow, however I don't understand why they can assume that $0<\alpha<1$ and why $q_n\leq (a_n+1)q_{n-1}$. Clearly I am missing something very obvious, but I can't see what is it.