Exercise 18, § 1

Let E be the space of sequences $\mathbf{x} = (x_n)_{n \in \mathbf{N}}$ of real numbers such that $\lim_{n \to \infty} x_n = 0$, endowed with the norm $\| \mathbf{x} \| = \sup_n |x_n|$, which is in fact a complete normed space. For each $\mathbf{x} = (x_n) \in E$ let $y = f(\mathbf{x})$ be the element $(y_n)$ of E defined by $y_n = |x_n|^{\frac{1}{2}} + \frac{1}{n+1}$; the function $f$ is continuous on E. Show that there is no solution of the differential equation $\mathbf{x}' = f(\mathbf{x})$ defined on a neighbourhood of 0 in $\mathbf{R}$, with values in E, and equal to 0 at the point $t = 0$ (cf. IV, p. 202, exerc. 11).