Show the language is decidable in context. Follow up from a previous question,

Let $S$ be a non-empty finite alphabet, Let $S^\ast$ denotes the set of words that can be written using the given alphabet. Let $A_1, A_2 \subseteq S*$ be languages such that membership in each of $A_1, A_2$ is decidable. Prove membership in the following language is decideable:
  $\{xy | x \in L_1, y \in L_2\}$

My initial guess is,
This a an appended string. 
How can I say this is "decidable?"

Given any word $w$, there are only finitely many ways to decompose it as $w=xy$, for words $x$ and $y$, and we may consider them each in turn in a computable manner. For each such decomposition $w=xy$, we can decide whether $x\in L_1$ and $y\in L_2$. If we find such an acceptable decomposition, then we accept $w$. If we don't, then we reject $w$ after all the possible decompositions are considered. In this way, we can computably decide membership in your set. 
