Why is every finite language A ⊆ Σ* regular

So I've been doing regular languages a while and still need a better understanding of why all finite languages A ⊆ Σ* are regular? Is there a formal proof of it or is it just because a DFA can represent any finite language since the states would be finite as well?

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.

StackExchange Computer Science Q#104322, answer score: 14