In the article: On $\omega$-limit sets of non-autonomous discrete systems by J. S. Canovas (for some it may hopefully appear as open access at https://www.tandfonline.com/doi/abs/10.1080/10236190500424274) at Proposition 1.4 the Author introduces a finite sequence of intervals $[a_i, b_i]$, which could be degenerate (points, that is). What is not clear to me is if the some of these intervals can be the same, that is if there can be $i\neq j$ such that $[a_i, b_i]=[a_j, b_j]$. In the proof this is not explicitly mentioned, but if my intuition is correct, following their construction these intervals should be allowed to also be non-distinct, but I am really not sure if some properties in the background may guarantee they actually are (which would make the statement stronger for applications). I would appreciate if anybody that can find this article could take a second to have a look and help me understand. I copy here the proposition in question, but most likely the whole proof is needed to get a full understanding. Note that $F$ and $P$ are the set of fixed points and periodic points, as customary.
Let $(f_{1,\infty})=(f_n)$ be a sequence of continuous interval maps which uniformly converges to a map $f$ such that $F(f^{2^m})=P(f)$ for some $m\in\mathbb{N}$. Then for any $x\in I$, $\omega(x,f_{1,\infty})=\bigcup_{i=1}^{2^m}[a_i,b_i]\subset F(f^{2^m})$. Moreover, $f([a_i,b_i])=[a_{i+1},b_{i+1}]$, $1\leq i\leq 2^m-1$, $f([a_{2^m},b_{2^m}])=[a_{1},b_{1}]$.
Thanks!
