Exercise 7

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

Define the sequence $\{ u_n \}$ by $$\begin{aligned} u_0(t) & = x_0, \\ u_{n+1} & = x_0 + \int_{t_0}^t f(s, u_n(s)) \, ds, \qquad n = 1, 2, \ldots. \end{aligned}$$ Use the result of the previous exercise to show that $$|f(t, u_n(t)) - f(t, u_{n-1}(t) )| \leq K|u_n(t) - u_{n-1}(t) |.$$