[proofplan] We sketch the three directions of the Dahlquist equivalence theorem: convergence implies consistency (the method must be exact for linear solutions), convergence implies the root condition (a violating root produces unbounded solutions of $y' = \mathbf{0}$), and consistency plus zero-stability implies convergence (via a discrete Gronwall argument on the error recurrence). [/proofplan] [step:Show convergence implies consistency] If the method converges, it must be exact for polynomial solutions $y(t) = a + bt$, which forces $\rho(1) = 0$ and $\rho'(1) = \sigma(1)$. These are equivalent to order $\geq 1$. [/step] [step:Show convergence implies the root condition] Apply the method to $y' = \mathbf{0}$, $y(0) = \mathbf{0}$. The iterates satisfy $\sum_\ell \rho_\ell y_{n+\ell} = 0$, a linear recurrence with general solution involving terms $n^j w_0^n$. If $|w_0| > 1$, the solution grows without bound. If $w_0$ is a repeated unit root, powers of $n$ appear, again causing unbounded growth. In either case, the numerical solution does not converge to $\mathbf{0}$, contradicting convergence. [/step] [step:Sketch sufficiency: consistency plus zero-stability implies convergence] The error $e_n = y_n - y(t_n)$ satisfies $\sum_\ell \rho_\ell e_{n+\ell} = h\sum_\ell \sigma_\ell[f(t_{n+\ell}, y_{n+\ell}) - f(t_{n+\ell}, y(t_{n+\ell}))] + \eta_{n+s}$, where $\eta_{n+s} = O(h^{p+1})$. The Lipschitz condition controls the first term. The root condition ensures the homogeneous recurrence has only bounded solutions. A discrete Gronwall-type argument using the resolvent of the recurrence yields $\max_n \|e_n\| \to 0$ as $h \to 0$. [/step]