# Prove that the language L = {w1, w1w2, w1w2w3, ..} is regular, provided wi is in a regular language

Let's assume that we're working over a finite alphabet $$\Sigma=\{a, b\}$$. How can one prove that $$L_2=\{w_1w_2...w_m| m ∈ \mathbb{N}, ∀i(w_i ∈ L)\}$$ is a regular language, provided that L is regular? By the way, L is a fixed language in the problem that I am solving, but it's regularity is trivial to prove so I have omitted it. This next step, however, I can't think of a valid way to prove. I would be glad if you guys could give me a hint.

## 1 Answer

Your language is known as $$L^+$$.

It is a very standard property of regular languages that if $$L$$ is regular then so is $$L^*$$; this can be proved in several ways, and you can find proofs in textbooks and online sources. Since $$L^+$$ is either equal to $$L^*$$ or obtained from it by removing $$\epsilon$$ (the empty string), then $$L^+$$ is also regular.

• Oh god. I was looking at some more complex constructions the whole day and been overthinking it. Thank you! – arnaudoff Jun 20 at 11:16