I am having hard time proving that the following language,comprised from two regular languages $L_1,L_2$(over the same $\Sigma$)is indeed regular:
$$L^\frown = \{ w\in \Sigma^* | w=u\sigma_1\mu_1...\sigma_n\mu_nv\}$$
$u,v \in \Sigma^*$
$0\leq n$, for every i($1\leq i \leq n$): $\sigma_i,\mu_i \in \Sigma$
$\sigma_1...\sigma_n \in L_1$
$\mu_1...\mu_n \in L_2$
I don't understand how to prove if because of the suffix and prefix(u,v accordingly).
If it weren't for u,v what I think I would've done is building an automaton(Deterministic finite automaton): $A=\left(Σ,Q1xQ2x\left\{1,2\right\},\left(q01,q02,2\right),F1xF2x2,δ\right)$, Ending in accepting iff both automatons that accept each language end in an accepting state, and $L_2$ needs to be after $L_1$ from the language's description. However, I don't know how to deal with the u,v in the beginning and in the end. I am not sure how to configure it correctly to prove $L^\frown$ is regular.
Would very much appreciate your assistance with it.