Theorem: Riemann's Rearrangement Theorem

Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $\R$ such that:
:$\ds \sum_{n \mathop = 1}^\infty a_n$ is conditionally convergent.
Let $\alpha, \beta \in \overline \R$ be such that:
:$-\infty \le \alpha \le \beta \le \infty$
Then there exists a permutation $\pi : \N \to \N$ such that, when we define:
:$\ds S_N = \sum_{n \mathop = 1}^N a_{\map \pi n}$
we have:
:$\ds \liminf_{N \mathop \to \infty} S_N = \alpha$ and $\ds \limsup_{N \mathop \to \infty} S_N = \beta$.
Setting $\alpha = \beta$, we can then say the following.
Let $S$ be a real infinite series which is conditionally convergent.
Then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges.

Proof:
Define the positive and negative parts:
:$a_n^+ = \max \set {0, a_n}$
and:
:$a_n^- = -\min \set {0, a_n}$
From Difference of Positive and Negative Parts, we have:
:$a_n = a_n^+ - a_n^-$
From Sum of Positive and Negative Parts, we have:
:$\size {a_n} = a_n^+ + a_n^-$ 
From Linear Combination of Convergent Series, we have:
:$\ds \sum_{n \mathop = 1}^\infty a_n^+ + \sum_{n \mathop = 1}^\infty a_n^- = \sum_{n \mathop = 1}^\infty \size {a_n} = \infty$
Hence we either have:
:$\ds \sum_{n \mathop = 1}^\infty a_n^+ = \infty$
or: 
:$\ds \sum_{n \mathop = 1}^\infty a_n^- = \infty$.
{{AimForCont}} only one of these sums are infinite.
We can write:
:$\ds \sum_{n \mathop = 1}^N a_n = \sum_{n \mathop = 1}^N a_n^+ - \sum_{n \mathop = 1}^N a_n^-$
Suppose that:
:$\ds \sum_{n \mathop = 1}^\infty a_n^+ = \infty$ and $\ds \sum_{n \mathop = 1}^\infty a_n^- < \infty$
Taking $N \to \infty$, we would obtain:
:$\ds \sum_{n \mathop = 1}^\infty a_n = \infty$
This contradicts that $\ds \sum_{n \mathop = 1}^\infty a_n$ is conditionally convergent
Suppose alternatively that:
:$\ds \sum_{n \mathop = 1}^\infty a_n^+ < \infty$ and $\ds \sum_{n \mathop = 1}^\infty a_n^- = \infty$
Taking $N \to \infty$, we would obtain:
:$\ds \sum_{n \mathop = 1}^\infty a_n = -\infty$
This again contradicts that $\ds \sum_{n \mathop = 1}^\infty a_n$ is conditionally convergent.
Hence we have:
:$\ds \sum_{n \mathop = 1}^\infty a_n^+ = \infty$
and:
:$\ds \sum_{n \mathop = 1}^\infty a_n^- = \infty$
Let $\sequence {\map \lambda n}_{n \mathop \in \N}$ be the indices $j$ for which $a_j \ge 0$, listed in increasing order.
Let $\sequence {\map \mu n}_{n \mathop \in \N}$ be the indices $j$ for which $a_j < 0$, listed in increasing order.
Let:
:$P_n = a_{\map \lambda n}$ for each $n \in \N$
and:
:$Q_n = -a_{\map \mu n}$ for each $n \in \N$.
Since:
:$\ds \sum_{n \mathop = 1}^\infty a_n^+ = \infty$
we have:
:$\ds \sum_{n \mathop = 1}^\infty P_n = \infty$
Since:
:$\ds \sum_{n \mathop = 1}^\infty a_n^- = \infty$
we have:
:$\ds \sum_{n \mathop = 1}^\infty Q_n = \infty$
Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a sequence converging to $\alpha$.
Let $\sequence {\beta_n}_{n \mathop \in \N}$ be a sequence such that $\beta_1 > 0$, $\alpha_n < \beta_n$ and $\beta_n \to \beta$.
Since:
:$\ds \sum_{n \mathop = 1}^\infty P_n = \infty$
there exists $N \in \N$ such that:
:$\ds \sum_{n \mathop = 1}^N P_n > \beta_1$
Let $m_1$ be the least such $N$. 
Now, since:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^\infty Q_n = -\infty$
there exists $N \in \N$ such that:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^N Q_n < \alpha_1$
Let $n_1$ be the least such $N$ so that:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n < \alpha_1$
Suppose now that we have constructed $n_1, \ldots, n_k \in \N$ and $m_1, \ldots, m_k \in \N$ such that:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{j - 1} + 1}^{m_j} P_n > \beta_j$
and:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{j - 1} + 1}^{m_j} P_n - \sum_{n \mathop = n_{j - 1} + 1}^{n_j} Q_n < \alpha_j$
for $1 \le j \le k$. 
We construct $n_{k + 1}$ and $m_{k + 1}$.
We have:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n + \sum_{n \mathop = m_k + 1}^\infty P_n = \infty$
Hence we can pick $m_{k + 1} \ge m_k + 1$ least such that:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n + \sum_{n \mathop = m_k + 1}^{m_{k + 1} } P_n > \beta_{k + 1}$
Then we have:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n + \sum_{n \mathop = m_k + 1}^{m_{k + 1} } P_n - \sum_{n \mathop = n_k + 1}^\infty Q_n = -\infty$
Hence we can pick $n_{k + 1} \ge n_k + 1$ least such that:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n + \sum_{n \mathop = m_k + 1}^{m_{k + 1} } P_n - \sum_{n \mathop = n_k + 1}^{n_{k + 1} } Q_n < \alpha_{k + 1}$
Hence the construction continues.
Define a permutation $\pi : \N \to \N$ by:
:$\map \pi j = \map \lambda j$ for $1 \le j \le m_1$
:$\map \pi {j + m_1} = \map \mu j$ for $1 \le j \le n_1$
:$\map \pi {j + m_k + n_k} = \map \lambda {j + m_k}$ for $1 \le j \le m_{k + 1} - m_k$ for $k \ge 1$
:$\map \pi {j + m_{k + 1} + n_k} = \map \mu {j + n_k}$ for $1 \le j \le n_{k + 1} - n_k$ for $k \ge 1$
Then we have:
:$\ds \sum_{j \mathop = 1}^{m_k + n_{k - 1} } a_{\map \pi j} = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n$
and:
:$\ds \sum_{j \mathop = 1}^{m_k + n_k} a_{\map \pi j} = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n$
First, set:
:$\ds x_k = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n$
and:
:$\ds y_k = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n$
We show that $x_k \to \beta$ and $y_k \to \alpha$.
By the choice of $m_k$ have:
:$\ds x_k = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n > \beta_k$
and:
:$\ds x_k - P_{m_k} = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k - 1} P_n \le \beta_k$
Hence we have:
:$x_k - P_{m_k} \le \beta_k < x_k$
Hence we have:
:$-P_{m_k} \le \beta_k - x_k < 0$
So:
:$\cmod {\beta_k - x_k} \le P_{m_k}$
From Terms in Convergent Series Converge to Zero, we have $a_j \to 0$ as $j \to \infty$.
Hence $P_{m_k} \to 0$ as $k \to \infty$ by Convergent Subsequence of Cauchy Sequence.
Since $\beta_k \to \beta$, it follows that $x_k \to \beta$ by the Squeeze Theorem.
We also have:
:$\ds y_k = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n < \alpha_k$
and:
:$\ds y_k + Q_{n_k} = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k - 1} Q_n \ge \alpha_k$
Hence we have:
:$y_k < \alpha_k \le y_k + Q_{n_k}$
so that:
:$0 < \alpha_k - y_k \le Q_{n_k}$
Hence:
:$\size {\alpha_k - y_k} \le Q_{n_k}$
We have $Q_{m_k} \to 0$ as $k \to \infty$ by Convergent Subsequence of Cauchy Sequence.
Since $\alpha_k \to \alpha$, we have that $y_k \to \alpha$ by the Squeeze Theorem.
Now set:
:$\ds S_N = \sum_{j \mathop = 1}^N a_{\map \pi j}$
For brevity set $n_0 = 0$. 
We have already shown that:
{{begin-eqn}}
{{eqn | l = \liminf_{k \mathop \to \infty} S_{m_k + n_k}
      | r = \liminf_{k \mathop \to \infty} y_k
      | rr= = \alpha
}}
{{eqn | l = \liminf_{k \mathop \to \infty} S_{m_k + n_{k - 1} }
      | r = \liminf_{k \mathop \to \infty} y_k
      | rr= = \alpha
}}
{{eqn | l = \limsup_{k \mathop \to \infty} S_{m_k + n_k}
      | r = \limsup_{k \mathop \to \infty} x_k
      | rr= = \beta
}}
{{eqn | l = \limsup_{k \mathop \to \infty} S_{m_k + n_{k - 1} }
      | r = \limsup_{k \mathop \to \infty} x_k
      | rr= = \beta
}}
{{end-eqn}}
by Convergence of Limsup and Liminf.
We consider two ranges of $N$:
:$m_k + n_{k - 1} < N < m_k + n_k$
:$m_k + n_k < N < m_{k + 1} + n_k$
for $k \ge 1$. 
To show that:
:$\ds \limsup_{N \mathop \to \infty} S_N = \beta$
it now suffices to show:
:$\ds \limsup_{\substack {N \mathop \to \infty \\ m_k + n_{k - 1} \mathop < N \mathop < m_k + n_k} } S_N \le \beta$
and:
:$\ds \limsup_{\substack {N \mathop \to \infty \\ m_k + n_k \mathop < N \mathop < m_{k + 1} + n_k} } S_N \le \beta$
by Limit Superior in terms of Partitioning Subsequences.
For $m_k + n_{k - 1} < N < m_k + n_k$, we have:
:$\ds S_N = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^N Q_n$
Write $k = \map k N$ to emphasize dependence on $N$.
Since $Q_n \ge 0$ for each $n \in \N$, we have:
:$\ds \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^N Q_n \le \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n = x_k$
That is, $S_N \le x_{\map k N}$. 
As $N \to \infty$, we have $\map k N \to \infty$, and hence by the Squeeze Theorem we have:
:$\ds \limsup_{\substack {N \mathop \to \infty \\ m_k + n_{k - 1} \mathop < N \mathop < m_k + n_k} } S_N \le \beta$
For $N \in \N$ with $m_k + n_k < N < m_{k + 1} + n_k$, we have:
:$\ds S_N = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^N P_n \le  \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n = x_k$
That is, $S_N \le x_{\map k N}$.
Taking $N \to \infty$, we get:
:$\ds \limsup_{\substack {N \mathop \to \infty \\ m_k + n_k \mathop < N \mathop < m_{k + 1} + n_k} } S_N \le \beta$
by the Squeeze Theorem. 
By Limit Superior in terms of Partitioning Subsequences, we obtain that:
:$\ds \limsup_{N \mathop \to \infty} S_N = \max \set {\limsup_{\substack {N \mathop \to \infty \\ m_k + n_{k - 1} \mathop < N \mathop < m_k + n_k} } S_N, \limsup_{\substack {N \mathop \to \infty \\ m_k + n_k \mathop < N \mathop < m_{k + 1} + n_k} } S_N, \limsup_{k \mathop \to \infty} S_{m_k + n_k}, \limsup_{k \mathop \to \infty} S_{m_{k + 1} + n_k} } = \beta$ 
To show that:
:$\ds \liminf_{N \mathop \to \infty} S_N = \alpha$
it now suffices to show:
:$\ds \liminf_{\substack {N \mathop \to \infty \\ m_k + n_{k - 1} \mathop < N \mathop < m_k + n_k} } S_N \ge \alpha$
and:
:$\ds \liminf_{\substack {N \mathop \to \infty \\ m_k + n_k \mathop < N \mathop < m_{k + 1} + n_k} } S_N \ge \alpha$
Again, for $m_k + n_{k - 1} < N < m_k + n_k$, we have:
:$\ds S_N = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^N Q_n$
Hence:
:$\ds S_N \ge \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots + \sum_{n \mathop = m_{k - 1} + 1}^{m_k} P_n - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n = y_k$
Hence:
:$\ds \liminf_{\substack {N \mathop \to \infty \\ m_k + n_{k - 1} \mathop < N \mathop < m_k + n_k} } S_N \ge \alpha$
since $y_k \to \alpha$, by Convergence of Limsup and Liminf and the Squeeze Theorem.
Also, for $m_k + n_k < N < m_{k + 1} + n_k$, we have:
:$\ds S_N = \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n + \sum_{n \mathop = m_k + 1}^N P_n \ge  \sum_{n \mathop = 1}^{m_1} P_n - \sum_{n \mathop = 1}^{n_1} Q_n + \sum_{n \mathop = m_1 + 1}^{m_2} P_n - \sum_{n \mathop = n_1 + 1}^{n_2} Q_n + \ldots - \sum_{n \mathop = n_{k - 1} + 1}^{n_k} Q_n = y_k$
Hence:
:$\ds \liminf_{\substack {N \mathop \to \infty \\ m_k + n_k \mathop < N \mathop < m_{k + 1} + n_k} } S_N \ge \alpha$
since $y_k \to \alpha$, by Convergence of Limsup and Liminf and the Squeeze Theorem.
We then have, by Limit Inferior in terms of Partitioning Subsequences:
:$\ds \liminf_{N \mathop \to \infty} S_N = \min \set {\liminf_{\substack {N \mathop \to \infty \\ m_k + n_{k - 1} \mathop < N \mathop < m_k + n_k} } S_N, \liminf_{\substack {N \mathop \to \infty \\ m_k + n_k \mathop < N \mathop < m_{k + 1} + n_k} } S_N, \liminf_{k \mathop \to \infty} S_{m_k + n_k}, \liminf_{k \mathop \to \infty} S_{m_{k + 1} + n_k} } = \alpha$ 
Hence we have found a permutation $\pi : \N \to \N$ such that when we define:
:$\ds S_N = \sum_{j \mathop = 1}^N a_{\map \pi j}$
we have:
:$\ds \liminf_{N \mathop \to \infty} S_N = \alpha$
and:
:$\ds \limsup_{N \mathop \to \infty} S_N = \beta$
If $\alpha = \beta$, then by Convergence of Limsup and Liminf, we have:
:$\ds \lim_{N \mathop \to \infty} S_N = \alpha$
and hence:
:$\ds \sum_{n \mathop = 1}^\infty x_{\map \pi n} = \alpha$ 
Hence we can find a permutation of $\ds \sum_{n \mathop = 1}^\infty x_n$ converging to any value in $\overline \R$.
{{qed}}