= Solution
Factor the first characteristic polynomial:
$$
\rho(\zeta)=(\zeta-1)(\zeta-5/7).
$$
The unit root is simple and the other root is strictly inside the unit disk, so the method is <zero-stable>. Also $\rho(1)=0$ and $\rho'(1)=\sigma(1)=2/7$, giving <numerical consistency>. The <Dahlquist equivalence theorem> therefore makes it convergent.
The mechanism can also be seen directly. The <zero-stable> homogeneous recurrence has bounded propagators because its two error modes are $1$ and $(5/7)^n$. Variation of constants for the perturbed recurrence bounds the error by a fixed multiple of its two starting errors and the sum of the step defects, plus $O(h)$ times a sum of errors from the Lipschitz vector field. Absorb the current implicit error term for sufficiently small $h$, then apply the <discrete Gronwall inequality>. On a fixed time interval the sum of $O(h^3)$ exact-start defects is $O(h^2)$, giving \b[second-order <numerical convergence> with second-order starting values]. The implicit step is locally well defined, for example by contraction when $4hL/7<1$ for a <Lipschitz constant> $L$.
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