How to prove the following related with regular languages

Question

How can we prove the following. If $$\sum$$ is any alphabet and L is any language $$L \subset \sum*$$ Then L*L* = L* ?

Answer 1

Since $\epsilon \in L^\star$, where $\epsilon$ is the empty word, it's clear that $L^\star L^\star \supset L^\star$.

Now, let $\omega \in L^\star L^\star$. Then $\omega = \omega_{1,1}\ldots\omega_{1,n}\omega_{2,1}\ldots\omega_{2,m}$ with all $\omega_{i,j} \in L$, by the definition of $\star$ and concatenation. But every word of the form $\omega'_1\ldots\omega'_k$, where all $\omega'_i \in L$, lies in $L^\star$... I'm leaving the rest to you.