Exercise 12, § 7

Let $(u_n)$ be a sequence of finite real numbers. For each permutation $\sigma$ of the set $\mathbf{N}$, put

a) If the series with general term $u_n$ is convergent, and if $\lim_{n \to \infty} r(n) = 0$, show that the series with general term $v_n = u_{\sigma(n)}$ is convergent to the same sum. [Consider the difference $\sum_{k=0}^n v_k - \sum_{k=0}^n u_k$ for large values of $n$, and if $h$ is the smallest integer $\leq n$ such that $\sigma(h) > n$, note that in this difference there are at most $\sigma(h) - h$ terms in each of the two sums which do not cancel out]. b) Suppose that the series whose general term is $u_n$ is convergent. Give an example of a permutation $\sigma$ such that $\lim_{n \to \infty} r(n) = 0$ but such that the criterion of Chapter III, § 5, Exercise 6 a) is not satisfied. [Define a family of mutually disjoint intervals $I_k = [n_k - 2k, n_k + 2k]$ in $\mathbf{N}$, and take $\sigma$ such that $\sigma(n) = n$ whenever $n$ does not belong to any of the $I_k$, and such that $\sigma(n_k - 2j) = n_k + 2j$, $\sigma(n_k + 2j) = n_k - 2j$ for all $k$ and $0 \leq j \leq k$].