Exercise 11

In Proof of the Existence and Uniqueness Theorem, prove the Existence and Uniqueness Theorem for first-order differential equations.

Since $$u_n(t) = u_1(t) + [u_2(t) - u_1(t)] + \cdots + [u_n(t) - u_{n-1}(t)],$$ we can view $u_n(t)$ as a partial sum for the series $$u_0(t) + \sum_{n=1}^\infty [u_n(t) - u_{n-1}(t)].$$ If we can show that this series converges absolutely, then our sequence will converge to a function $u(t)$. Show that $$\sum_{n=1}^\infty |u_n(t) - u_{n-1}(t)| \leq \frac{M}{K} \sum_{n=1}^\infty \frac{(K |t - t_0|)^n}{n!} \leq \frac{M}{K} \left( e^{K|h|} - 1 \right),$$ where $h$ is the maximum distance between $(t_0, x_0)$ and the boundary of the rectangle $R$. Since $|u_n(t) - u_{n -1}(t)| \to 0$, we know that $u_n(t)$ converges to a continuous function $u(t)$ that solves our equation.