If none of the languages $L_1$,$L_2$,... is regular, and $L_i \subseteq L_{i+1}$ for each i, is $\bigcup_{n=1}^\infty L_i$ regular?
I guess the answer is no for any given languages, but I cannot formulate my arguments properly.
Thank you for the help.
Let $L$ be a language that is not regular. Let $s$ be any string not in $L$. I claim that the language $L \cup \{ s \} $ is also not regular.
By adding in all strings not in $L$ one at a time, you get a nested union of non-regular languages whose union consists of all strings, and is therefore regular.
So no, you cannot conclude that the nested union of non-regular languages is also non-regular.