In the book of Analysis On Manifolds by Munkres, at page 141, it is given that
However, I think there is a problem in the existence such an $M$.
Because we do know that the supports of $\phi_i$s, i.e $S_i$s, will form a cover for $A$, and hence for $D$, but by the compactness of $D$, we can find a finitely many $S_i$s that will cover $D$, but this does not mean that $D$ has a non-empty intersection with only finitely many $S_i$s. Therefore, there might be cases where $D$ have intersect with infinitely many $S_i$s, so the existence of $M$ is not clear, IMO.
I should point out also that the very definition of $\phi$s requires that for any $x\in A$, there exists an neighbourhood of $x$ s.t only finitely many of $S_i$ will intersect with that nbd.However, $D$ possibly have infinitely many element, so I can't see how can one derive that from the local finiteness the existence of $M$.