Is infinite union of non-regular languages $L_i$ that form a chain such that $L_i\subseteq L_{i+1}$ always non-regular?
Or is there a possibility that it be ever regular?
Is there an easy way to see this?
Is infinite union of non-regular languages $L_i$ that form a chain such that $L_i\subseteq L_{i+1}$ always non-regular?
Or is there a possibility that it be ever regular?
Is there an easy way to see this?
Let $L$ be a non-regular language over $\Sigma$, and let $w_1,w_2,\ldots$ be an enumeration of $\Sigma^*\setminus L$. Define $L_i = L \cup \{ w_j : j \geq i \}$. Each $L_i$ is non-regular (why?), $L_i \subset L_{i+1}$, and $\bigcup_i L_i = \Sigma^*$.