(a) A numerical method for solving the ordinary differential equation

$$y^{\prime}(t)=f(t, y), \quad t \in[0, T], \quad y(0)=y_{0},$$

generates for every $h>0$ a sequence $\left\{y_{n}\right\}$, where $y_{n}$ is an approximation to $y\left(t_{n}\right)$ and $t_{n}=n h$. Explain what is meant by the convergence of the method.

(b) Prove from first principles that if the function $f$ is sufficiently smooth and satisfies the Lipschitz condition

$$|f(t, x)-f(t, y)| \leqslant \lambda|x-y|, \quad x, y \in \mathbb{R}, \quad t \in[0, T],$$

for some $\lambda>0$, then the trapezoidal rule

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

converges.