Why is every finite language A ⊆ Σ* regular
The proof goes something like this:


If $A$ is a finite language, then it contains a finite number of strings $a_0, a_1, \cdots , a_n$.
The language $\{a_i\}$ consisting of a single literal string $a_i$ is regular.
The union of a finite number of regular languages is also regular.
Therefore, $A = \{a_0\} \cup \{a_1\} \cup \cdots \cup \{a_n\}$ is regular.