TITLE: Convergence of a rearrangement of a series
QUESTION [7 upvotes]: Let $\sum a_n$ be a convergent series and $f$ is a bijection on $\mathbb{N}$.

Suppose $(f(n) -n)$ is a bounded sequence the rearrangement series $\sum a_{f(n)}$ converges to same limit.

Suppose $m_n=\sup\{|a_k| : k> n\}$. If the sequence $(m_n |f(n) -n|) $ converges to $0$, then the rearrangement series converges to same limit.


Since $(f(n) -n)$ is bounded sequence, the terms can be grouped in some way such that the limit does not change. But how it can be done. $2$ seems consequence of $1$. How to do it?

REPLY [3 votes]: 2 Let
$$\begin{align*}
x_n= |S_n - S'_n| &= |a_1+\cdots+ a_n -(a_{f(1)}+\cdots+a_{f(n)})| \\
&= \left| \sum_{i\in C_n}a_i + \sum_{i\in D_n}a_i - \sum_{i\in A_n}a_{f(i)} - \sum_{i\in B_n}a_{f(i)} \right|
\end{align*},$$
where $A_n=\{i\in\mathbb N: f(i)\leq n, 1\leq i\leq n\}$, $B_n=\{i\in\mathbb N: f(i)> n, 1\leq i\leq n\}$, $C_n =\{i\in\mathbb N: i=f(j), j\in A_n \}$ and $D_n=\{1,\dots, n \} \setminus  C_n$. Then we have
$$\sum_{i\in C_n}a_i - \sum_{i\in A_n}a_{f(i)} = 0. \quad\quad(1)$$
Now, becouse for each $k\in\mathbb N$ there is $l\in\mathbb N$ such that
$$\forall_{i\leq k} \exists_{j\leq l} \quad i=f(j)$$
thus $h(n):=\min\{i:i\in D_n\} \to \infty$ when $n\to\infty$.
We denote $p(n) :=\max\{|f(n)-n|: n\in B_n\}=|f(k_n) - k_n|$ for some $k_n\in\mathbb N:h(n)\leq k_n\leq n$ and then we have
$$N(D_n)=N(B_n)\leq p(n) \quad\quad (2)$$
where $N(X)$ is the number of elements of $X$.
Let $\epsilon > 0$. Then... [what then?]