Exercise 15, § 5

Let $(x_n)$ be a sequence of finite real numbers and let $(p_n)$ be a sequence of finite real numbers $\geq 0$ such that $\lim_{n \to \infty} \left( \sum_{i=0}^n p_i \right) = +\infty$. Let $y_n = \left( \sum_{i=0}^n p_i x_i \right) / \left( \sum_{i=0}^n p_i \right)$ for those values of $n$ such that $\sum_{i=1}^n p_i \neq 0$. Show that $\liminf_{n \to \infty} x_n \leq \liminf_{n \to \infty} y_n \leq \limsup_{n \to \infty} y_n \leq \limsup_{n \to \infty} x_n$.

Let $H$ be any non-empty subset of the set of cluster points of the sequence $(x_n)$ in $\overline{\mathbf{R}}$. Show that the sequence $(p_n)$ of finite real numbers $\geq 0$ can be determined so that $\lim_{n \to \infty} \left( \sum_{i=0}^n p_i \right) = +\infty$ and so that the set of cluster points of the corresponding sequence $(y_n)$ contains $H$. [Reduce to the case where $H$ is countable, then define $(p_n)$ by induction, taking its terms to be 0 or 1].

Deduce that the sequence $(x_n)$ converges in $\overline{\mathbf{R}}$ if and only if the sequence $(y_n)$ converges in $\overline{\mathbf{R}}$ for every sequence $(p_n)$ of numbers $\geq 0$ such that $\lim_{n \to \infty} \left( \sum_{i=0}^n p_i \right) = +\infty$.