= Solution
The first characteristic polynomial of the <linear multistep method> comes only from its left side:
$$
\rho(\zeta)=\zeta^2-\zeta=\zeta(\zeta-1).
$$
Its roots are zero and one; both lie in the closed unit disc, and the only unit-modulus root is simple. Thus the method satisfies the <root condition for a multistep method> and is <zero-stability>[zero-stable]. Part (b) showed consistency for every $\alpha,\beta$. The <Dahlquist equivalence theorem> states that a consistent linear multistep method is convergent exactly when it is zero-stable. Therefore
$$
\boxed{\text{the method is convergent}}
$$
under the usual Lipschitz hypotheses on $f$.
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