(i) Explain briefly what is meant by the convergence of a numerical method for ordinary differential equations.

(ii) Suppose the sufficiently-smooth function $f: \mathbb{R} \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ obeys the Lipschitz condition: there exists $\lambda>0$ such that

$$\|\mathbf{f}(t, \mathbf{x})-\mathbf{f}(t, \mathbf{y})\| \leqslant \lambda\|\mathbf{x}-\mathbf{y}\|, \quad \mathbf{x}, \mathbf{y} \in \mathbb{R}^{d}, t \geqslant 0$$

Prove from first principles, without using the Dahlquist equivalence theorem, that the trapezoidal rule

$$\mathbf{y}_{n+1}=\mathbf{y}_{n}+\frac{1}{2} h\left[\mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)+\mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)\right]$$

for the solution of the ordinary differential equation

$$\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y}), \quad t \geqslant 0, \quad \mathbf{y}(0)=\mathbf{y}_{0},$$

converges.