regular language proof

Question:

Prove that a finite language is a regular language.

How would I go about solving this? I tried my own approach (below) but didn't get far because I don't understand how I am supposed to approach this proof.

My Approach:

Consider a finite automata M

($$\delta$$* denotes $$\delta$$ hat(^) which means all states w travels to until empty)

$$M=(Q,\Sigma,\delta,q_0,F)$$

if w is a string and w $$\in$$L* (any finite language)

=> $$\delta$$*($$q_0$$, w)$$\in$$F (then i make a logic statement that says for all w this is true?)

and if w$$\in\epsilon$$

=> $$\delta$$*($$q_0$$, $$\epsilon$$ )$$\in$$F

Let $$n$$ be the maximum length of the words in the given finite language $$L$$, and let $$m$$ be the size of the (occuring letters from the) alphabet.
Build an automata with $$\sum_{k\le n}m^k$$ many states, one for each possible word, and mark exactly those states as accepting which correspond to a word in $$L$$.

• you mean draw new DFA?would emphasis your prove please. – Jared Feb 19 at 0:13