How to reduce language is finite to language is regular? I'm trying to reduce $FIN \leq REGU$, where:

$$FIN = \{ \langle M \rangle \vert T(M) &lt; \infty \} \\ REGU  = \{\langle M \rangle \vert T(M) \in REG\}$$

Now one thing I have done so far is this: 

$ w \in FIN \implies | T(M_w) &lt; \infty \implies \exists n \in N: L(M_w) = \{w_0, \ldots, w_n \} \implies \exists  \ \text{DFA} \  A: L(A) = L(M_w)  $

What I want to show now that : "to decide if a language is regular is at least as hard as deciding wether it is finite"

Thats where I'm stuck. 



I want to remark that this is not homework but, but a exam preparation question.
First note, that for a regular language, however encoded, it is decidable wether it is finite or not. So you can use the language $REGU$ as an oracle for $FIN$ in the following way: To check if $\langle M \rangle \in FIN$ you ask the oracle $\langle M \rangle \in REGU$? If the answer is no, you can be sure that $\langle M \rangle \notin FIN$ as every non-regular language is infinite. If the answer is yes there is a proof (e.g. an NFA $\mathcal{A}$) that shows that $M$'s language is regular. Then you run your procedure to check whether $\mathcal{A}$ is finite (this is, as mentioned above decidable).