Exercise 24, § 2

a) Let $(a_k)_{1 \leq k \leq n}$ be a finite sequence of points in a normed space $E$, and let $(\lambda_k)_{1 \leq k \leq n}$ be a finite sequence of numbers $> 0$ such that $\sum_{k=1}^n \lambda_k < 1$; put $\mu_k = 1 - \sum_{j=1}^{k-1} \lambda_j$ for every $k$; then

b) Let $(a_n)$ be an infinite sequence of points in the unit ball of $E$, and let $(\lambda_n)$ be an infinite sequence of numbers $> 0$ such that $\sum \lambda_n = 1$. For every $n > 0$, put $\mu_n = 1 - \sum_{k=1}^{n-1} \lambda_k$ and $b_n = \sum_{k=1}^{n-1} \lambda_k a_k + \mu_n a_n$; show that, for all $n \geq 1$, we have (Apply a) by induction.) c) Let $(C_n)$ be a decreasing sequence of convex sets in $E$, contained in the unit ball, and suppose that $d(0, C_1) \geq \theta > 0$ (hence, a fortiori $d(0, C_n) \geq \theta$ for all $n$). Let $(\lambda_n)$ be a sequence of numbers $> 0$ such that $\sum \lambda_n = 1$. Show that there exists a number $\alpha$ such that $\theta \leq \alpha \leq 1$ and a sequence $(x_n)$ of points of $E$ such that $x_n \in C_n$ for all $n$, $\|\sum_n \lambda_n x_n\| = \alpha$ and, for all $n$

(Take $x_1$ such that $\|x_1\|$ is arbitrarily close to $d(0, C_1)$, then, by induction, take $x_n$ such that $\left\| \sum_{k=1}^{n-1} \lambda_k x_k + (\sum_{j=n}^\infty \lambda_j) x_n \right\|$ is arbitrarily close to the infimum of the numbers $\left\| \sum_{k=1}^{n-1} \lambda_k x_k + (\sum_{j=n}^\infty \lambda_j) y \right\|$ where $y$ ranges over $C_n$. Then use b).)