Past exam of the numerical analysis course of the University of Oxford / 2021 / 3 / c / iii / Solution

Yes: for $k=2$ the stated inequality guarantees zero-stability. Part (ii) puts every polynomial root in the closed unit disc. The remaining root condition for a multistep method is that every root on the unit circle has multiplicity of a root one.

If this failed for a quadratic polynomial, it would have a repeated root $\xi$ with $|\xi |=1$, andConsequently $|{\alpha }_1|=2|{\alpha }_2|$ and $|{\alpha }_0|=|{\alpha }_2|$, contradictingbecause ${\alpha }_2\ne 0$. Thus all roots on the unit circle are simple andThis conclusion does not assert consistency of a numerical method: part (i) rules out consistency of a numerical method whenever the coefficient inequality is strict. Zero-stability and consistency of a numerical method are separate requirements.