Exercise 1, § 5

Suppose that $K = \mathbf{R}$. Let $r$ be an integer $\geq 1$ or $\infty$. A group of class $C^r$ is a set $G$ with a group structure and a manifold structure of class $C^r$ such that the mapping $(x, y) \mapsto xy^{-1}$ of $G \times G$ into $G$ is of class $C^r$.

(a) Suppose henceforth that $r \geq 2$. A neighbourhood of $e$ in $G$ is identified with a neighbourhood of $0$ in a Banach space. Let $xy = P(x, y)$, where $P$ is a mapping of class $C^2$ on an open neighbourhood of $(0, 0)$. Let

Show that

as $(x, y) \to (0, 0)$. (Use an expansion of P of order 2 with integral remainder.) (b) Show that

as $(x, y)$ tends to $(0, 0)$.