Exercise 10, § 4

Let $G$ be a Lie group over $K$. Prove that there exists a neighbourhood $V$ of $e$ possessing the following property: for every sequence $(x_n)$ of elements of $V$, if one defines recursively $y_1 = x_1, y_n = (x_n, y_{n-1})$, the sequence $(y_n)$ tends to $e$.