Prove that the Euro system $c = (200, 100, 50, 20, 10, 5, 2, 1)$ is canonical.
To do this I am using Definition 2 and Theorem 2 from https://arxiv.org/pdf/0809.0400.pdf which state:
Definition 2: A coin system \$ is non-canonical if there is an $x$ with $\vert GRD_\$(x) \vert $ $>$ $\vert OPT_\$(x) \vert$, and such $x$ is called a counterexample of \$.
Theorem 2: Let $\$_1 = \langle c_m, c_{m-1}, · · · , c_2, 1 \rangle$ and $\$_2 = \langle c_{m+1}, c_m, · · · , c_2, 1\rangle $ be two coin systems such that $\$_1$ is canonical but $\$_2$ is not. Then there is some $k$ such that $k·c_m < c_{m+1} < (k+1)·c_m$ and $(k+1)·c_m$ is a counterexample of $\$_2$.
I attempt this proof by choosing $\$_1 = (100, 50, 20, 10, 5, 2, 1)$ and $\$_2=c$.
Assuming by contradiction that $c$ is non-canonical then Theorem 2 leads us to finding $k=3$ and therefore our counterexample would be $x=400$.
In this case, $GRD_\$(400)=(2,0,0,0,0,0,0,0)$ and $OPT_\$(400)=(2,0,0,0,0,0,0,0)$ which is a contradiction to Definition 2 so $c$ is canonical.
Question
My question is, how can I easily show that my chosen $\$_1$ is canonical. Furthermore, is there a way to prove Theorem 2?