= Solution
At $h=0$ the first characteristic polynomial is
$$
\rho(\xi)=\xi^2-\frac45\xi-\frac15
=(\xi-1)\left(\xi+\frac15\right).
$$
Its roots satisfy the <root condition for a multistep method>: the only unit-modulus root is the simple root $1$, and the other root has modulus $1/5$. The method is consistent because part a gives positive order. The <Dahlquist equivalence theorem>, in the form allowed by the question for this derivative-augmented method, therefore proves convergence.
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