Summable series and the Riemann rearrangement theorem
1 Introduction
Let $\mathbb{N}$ be the set of positive integers. A function from $\mathbb{N}$ to a set is called a sequence. If $X$ is a topological space and $x\in X$, a sequence $a:\mathbb{N}\to X$ is said to converge to $x$ if for every open neighborhood $U$ of $x$ there is some ${N}_{U}$ such that $n\ge {N}_{U}$ implies that ${a}_{n}\in U$. If there is no $x\in X$ for which $a$ converges to $x$, we say that $a$ diverges.
Let $a:\mathbb{N}\to \mathbb{R}$. We define $s(a):\mathbb{N}\to \mathbb{R}$ by ${s}_{n}(a)={\sum}_{k=1}^{n}{a}_{k}$. We call ${s}_{n}(a)$ the $n$th partial sum of the sequence $a$, and we call the sequence $s(a)$ a series. If there is some $\sigma \in \mathbb{R}$ such that $s(a)$ converges to $\sigma $, we write
$$\sum _{k=1}^{\mathrm{\infty}}{a}_{k}=\sigma .$$ 
2 Goldbach
Euler [22, §110]: “If, as is commonly the case, we take the sum of a series to be the aggregate of all of its terms, actually taken together, then there is no doubt that only infinite series that converge continually closer to some value, the more terms we actually add, can have sums”.
Euler Goldbach correspondence nos. 55, 161, 162.
3 Dirichlet
In 1837 Dirichlet proved that one can rearrange terms in an absolutely convergent series and not change the sum, and gave examples to show that this was not the case for conditionally convergent series.
If $a$ is a sequence and the series $s(a)$ converges, we say that the series $s(a)$ is absolutely convergent. Because $\mathbb{R}$ is a complete metric space, a series being absolutely convergent implies that it is convergent.
In the following theorem we prove that if a series converges absolutely, then every rearrangement of it converges to the same value. Our proof follows Landau [45, p. 157, Theorem 216].
Theorem 1.
If $a$ is a sequence for which $s\mathit{}\mathrm{(}a\mathrm{)}$ converges absolutely and
$$\sum _{n=1}^{\mathrm{\infty}}{a}_{n}=\sigma ,$$ 
then for any bijection $\lambda \mathrm{:}\mathrm{N}\mathrm{\to}\mathrm{N}$, the series $s\mathit{}\mathrm{(}a\mathrm{\circ}\lambda \mathrm{)}$ converges to $\sigma $.
Proof.
Let $\u03f5>0$, and let $M$ be large enough so that
$$ 
Let $r$ be large enough so that
$$ 
Fix $m\ge r$, and let $h:\mathbb{N}\to \mathbb{N}$ be the sequence whose terms are the elements of
$$\mathbb{N}\setminus \{{\lambda}_{n}:1\le n\le m\}$$ 
arranged in ascending order. If $t+m\ge {\mathrm{max}}_{1\le n\le m}{\lambda}_{n}$ then
$$\{{\lambda}_{n}:1\le n\le m\}\cup \{{h}_{n}:1\le n\le t\}=\{n:1\le n\le t+m\},$$ 
and hence
$$\sum _{n=1}^{m}{a}_{{\lambda}_{n}}+\sum _{n=1}^{t}{a}_{{h}_{n}}=\sum _{n=1}^{t+m}{a}_{n}.$$ 
Taking $t\to \mathrm{\infty}$, we get
$$\sum _{n=1}^{m}{a}_{{\lambda}_{n}}+\sum _{n=1}^{\mathrm{\infty}}{a}_{{h}_{n}}=\sigma ;$$ 
the series $s(a\circ h)$ converges because for sufficiently large $n$, ${h}_{n}=n$. Hence, for every $m\ge r$,
$$ 
which shows that $s(a\circ \lambda )$ converges to $\sigma $. ∎
4 Riemann rearrangement theorem
If $a:\mathbb{N}\to \mathbb{R}$ and $\lambda :\mathbb{N}\to \mathbb{N}$ is a bijection, we call the sequence $a\circ \lambda :\mathbb{N}\to \mathbb{R}$ a rearrangement of the sequence $a$.
Because $\mathbb{N}$ is a wellordered set, if there are at least $n$ elements in the set $\{k\in \mathbb{N}:{a}_{k}\ge 0\}$ then it makes sense to talk about the $n$th nonnegative term in the sequence $a$. If $a$ were not a function from $\mathbb{N}$ to $\mathbb{R}$ but merely a function from a countable set to $\mathbb{R}$, it would not make sense to talk about the $n$th nonnegative term in $a$ or the $n$th negative term in $a$.
Riemann [60, pp. 9697]
Our proof follows Landau [45, p. 158, Theorem 217].
Theorem 2 (Riemann rearrangement theorem).
If $a\mathrm{:}\mathrm{N}\mathrm{\to}\mathrm{R}$ and $s\mathit{}\mathrm{(}a\mathrm{)}$ converges but $s\mathit{}\mathrm{(}\mathrm{}a\mathrm{}\mathrm{)}$ diverges, then for any nonnegative real number $\sigma $ there is some rearrangement $b$ of $a$ such that $s\mathit{}\mathrm{(}b\mathrm{)}\mathrm{\to}\sigma $.
Proof.
Define $p,q:\mathbb{N}\to \mathbb{R}$ by
$${p}_{n}=\frac{{a}_{n}+{a}_{n}}{2},{q}_{n}=\frac{{a}_{n}{a}_{n}}{2}.$$ 
${p}_{n}$ and ${q}_{n}$ are nonnegative, and satisfy ${p}_{n}{q}_{n}={a}_{n}$, ${p}_{n}+{q}_{n}={a}_{n}$. If one of $s(p)$ or $s(q)$ converges and the other diverges, we obtain a contradiction from
$${s}_{n}(a)=\sum _{k=1}^{n}{a}_{k}=\sum _{k=1}^{n}({p}_{k}{q}_{k})=\sum _{k=1}^{n}{p}_{k}\sum _{k=1}^{n}{q}_{k}={s}_{n}(p){s}_{n}(q)$$ 
and the fact that $s(a)$ converges. If both $s(p)$ and $s(q)$ converge, then we obtain a contradiction from
$${s}_{n}(a)=\sum _{k=1}^{n}{a}_{k}=\sum _{k=1}^{n}({p}_{k}+{q}_{k})=\sum _{k=1}^{n}{p}_{k}+\sum _{k=1}^{n}{q}_{k}={s}_{n}(p)+{s}_{n}(q)$$ 
and the fact that $s(a)$ diverges. Therefore, both $s(p)$ and $s(q)$ diverge.
Because $s(a)$ converges and $s(a)$ diverges, there are infinitely many $n$ with ${a}_{n}>0$ and there are infinitely many $n$ with $$. Let ${P}_{n}$ be the $n$th nonnegative term in the sequence $a$, and let ${Q}_{n}$ be the absolute value of the $n$th negative term in the sequence $a$. The fact that $s(p)$ diverges implies that $s(P)$ diverges, and the fact that $s(q)$ diverges implies that $s(Q)$ diverges.
Let $\sigma \ge 0$. We define sequences $\mu ,\nu :\mathbb{N}\to \mathbb{N}$ by induction as follows. Let ${\mu}_{1}$ be the least element of $\mathbb{N}$ such that
$${s}_{{\mu}_{1}}(P)>\sigma ,$$ 
and with ${\mu}_{1}$ chosen, let ${\nu}_{1}$ be the least element of $\mathbb{N}$ such that
$$ 
Let ${m}_{2}$ be the least element of $\mathbb{N}$ such that
$${s}_{{\mu}_{2}}(P){s}_{{\nu}_{1}}(Q)>\sigma ,$$ 
and with ${\mu}_{2}$ chosen, let ${\nu}_{2}$ be the least element of $\mathbb{N}$ such that
$$ 
It is straightforward to check that ${\mu}_{2}>{\mu}_{1}$ and ${\nu}_{2}>{\nu}_{1}$.
Suppose that ${\mu}_{1},\mathrm{\dots},{\mu}_{n}$ and ${\nu}_{1},\mathrm{\dots},{\nu}_{n}$ have been chosen, that ${\mu}_{n}$ is the least element of $\mathbb{N}$ such that
$${s}_{{\mu}_{n}}(P){s}_{{\nu}_{n1}}(Q)>\sigma ,$$ 
that ${\nu}_{n}$ it the least element of $\mathbb{N}$ such that
$$ 
and that ${\mu}_{n}>{\mu}_{n1}$ and ${\nu}_{n}>{\nu}_{n1}$. Let ${\mu}_{n+1}$ be the least element of $\mathbb{N}$ such that
$${s}_{{\mu}_{n+1}}(P){s}_{{\nu}_{n}}(Q)>\sigma ,$$ 
and with ${\mu}_{n+1}$ chosen, let ${\nu}_{n+1}$ be the least element of $\mathbb{N}$ such that
$$ 
It is straightforward to check that ${\mu}_{n+1}>{\mu}_{n}$ and ${\nu}_{n+1}>{\nu}_{n}$.
Define $b:\mathbb{N}\to \mathbb{R}$ by taking ${b}_{n}$ to be the $n$th term in
$${P}_{1},\mathrm{\dots},{P}_{{\mu}_{1}},{Q}_{1},\mathrm{\dots},{Q}_{{\nu}_{1}},{P}_{{\mu}_{1}+1},\mathrm{\dots},{P}_{{\mu}_{2}},{Q}_{{\nu}_{1}+1},\mathrm{\dots},{Q}_{{\nu}_{2}},\mathrm{\dots},$$ 
which, because the sequences $\mu $ and $\nu $ are strictly increasing, is a rearrangement of the sequence $a$.
∎
5 Symmetry
Don’t use order where it is accidental.
6 Nets
A directed set is a set $D$ and a binary relation $\u2aaf$ satisfying

•
if $m,n,p\in D$, $m\u2aafn$, and $n\u2aafp$, then $m\u2aafp$

•
if $m\in D$, then $m\u2aafm$

•
if $m,n\in D$, then there is some $p\in D$ such that $m\u2aafp$ and $n\u2aafp$.
For example, let $A$ be a set, let $D$ be the set of all subsets of $A$, and say that $F\u2aafG$ when $F\subseteq G$. Check that $(D,\u2aaf)$ is a directed set: for $F,G\in D$, we have $F\cup G\in D$, and $F\cup G$ is an upper bound for both $F$ and $G$.
A net is a function from a directed set $(D,\u2aaf)$ to a set $X$. Let $(X,\tau )$ be a topological space, let $S:(D,\u2aaf)\to (X,\tau )$ be a net, and let $x\in X$. We say that $S$ converges to $x$ if for every $U\in \tau $ with $x\in U$ there is some ${N}_{U}\in D$ such that ${N}_{U}\u2aafi$ implies that $S(i)\in U$. One proves that a topological space is Hausdorff if and only if every net in this space converges to at most one point [41, p. 67, Theorem 3].
A net $S:(D,\u2aaf)\to \mathbb{R}$ is said to be increasing if $m\u2aafn$ implies that $S(m)\le S(n)$.
Lemma 3.
If $S\mathrm{:}\mathrm{(}D\mathrm{,}\mathrm{\u2aaf}\mathrm{)}\mathrm{\to}\mathrm{R}$ is an increasing net and the range $R$ of $S$ has an upper bound, then $S$ converges to the supremum of $R$.
Proof.
Because $R$ is a subset of $\mathbb{R}$ that has an upper bound, it has a supremum, call it $\sigma $. To say that $\sigma $ is the supremum of $R$ means that for all $r\in R$ we have $r\le \sigma $ ($\sigma $ is an upper bound) and that for all $\u03f5>0$ there is some ${r}_{\u03f5}\in R$ with $$ (nothing less than $\sigma $ is an upper bound). Take $\u03f5>0$. There is some ${r}_{\u03f5}\in R$ with $$. As ${r}_{\u03f5}\in R$, there is some ${n}_{\u03f5}\in D$ with $S({n}_{\u03f5})={r}_{\u03f5}$. If ${n}_{\u03f5}\u2aafn$, then because $S$ is increasing, $S({n}_{\u03f5})\le S(n)$, and hence
$$ 
But $S(n)\in R$, so $S(n)\le \sigma $. Hence ${n}_{\u03f5}\u2aafn$ implies that $$, showing that $S$ converges to $\sigma $. ∎
7 Unordered sums
Let $A$ be a set, and let ${\mathcal{P}}_{0}(A)$ be the set of all finite subsets of $A$. Check that $({\mathcal{P}}_{0}(A),\subseteq )$ is a directed set: if $F,G\in {\mathcal{P}}_{0}(A)$ then $F\cup G\in {\mathcal{P}}_{0}(A)$ and $F\cup G$ is an upper bound for both $F$ and $G$. Let $f:A\to \mathbb{R}$ be a function, and define ${S}_{f}:{\mathcal{P}}_{0}(A)\to \mathbb{R}$ by
$${S}_{f}(F)=\sum _{a\in F}f(a),F\in {\mathcal{P}}_{0}(A).$$ 
If the net ${S}_{f}$ converges, we say that the function $f$ is summable, and we call the element of $\mathbb{R}$ to which ${S}_{f}$ converges the unordered sum of $f$, denoted by
$$\sum _{a\in A}f(a).$$ 
If $B$ is a subset of $A$, we say that $f$ is summable over $B$ if the restriction of $f$ to $B$ is summable. If ${f}_{B}$ is the restriction of $f$ to $B$ and $f$ is summable over $B$ (i.e. ${f}_{B}$ is summable), by
$$\sum _{a\in B}f(a)$$ 
we mean
$$\sum _{a\in B}{f}_{B}(a).$$ 
Lemma 4.
Suppose that $f\mathrm{,}g\mathrm{:}A\mathrm{\to}\mathrm{R}$ are functions and $\alpha \mathrm{,}\beta \mathrm{\in}\mathrm{R}$. If $f$ and $g$ are summable, then $\alpha \mathit{}f\mathrm{+}\beta \mathit{}g$ is summable and
$$\sum _{a\in A}(\alpha f(a)+g(a))=\alpha \sum _{a\in A}f(a)+\beta \sum _{a\in A}g(a).$$ 
Proof.
Let ${\sigma}_{1}={\sum}_{a\in A}f(a)$ and ${\sigma}_{2}={\sum}_{a\in A}g(a)$, and set $h=\alpha f+\beta g$. For $\u03f5>0$, there is some ${F}_{\u03f5}\in {\mathcal{P}}_{0}(A)$ such that ${F}_{\u03f5}\subseteq F\in {\mathcal{P}}_{0}(A)$ implies that $$, and there is some ${G}_{\u03f5}\in {\mathcal{P}}_{0}(A)$ such that ${G}_{\u03f5}\subseteq G\in {\mathcal{P}}_{0}(A)$ implies that $$. Let ${H}_{\u03f5}={F}_{\u03f5}\cup {G}_{\u03f5}\in {\mathcal{P}}_{0}(A)$. If ${H}_{\u03f5}\subseteq H\in {\mathcal{P}}_{0}(A)$, then, as ${F}_{\u03f5}\subseteq H$ and ${G}_{\u03f5}\subseteq H$,
${S}_{h}(H)(\alpha {\sigma}_{1}+\beta {\sigma}_{2})$  $=$  $\left{\displaystyle \sum _{a\in H}}(\alpha f(a)+\beta g(a))\alpha {\sigma}_{1}\beta {\sigma}_{2}\right$  
$=$  $\alpha {S}_{f}(H)+\beta {S}_{g}(H)\alpha {\sigma}_{1}\beta {\sigma}_{2}$  
$\le $  $\alpha {S}_{f}(H){\sigma}_{1}+\beta {S}_{g}(H){\sigma}_{2}$  
$\le $  $\alpha \u03f5+\beta \u03f5;$ 
we write $\le $ rather than $$ in the last inequality to cover the case where $\alpha =\beta =0$. It follows that ${S}_{h}$ converges to $\alpha {\sigma}_{1}+\beta {\sigma}_{2}$. ∎
The following lemma is simple to prove and ought to be true, but should not to be called obvious. For example, the Cesàro sum of the sequence $1,1,1,1,\mathrm{\dots}$ is $\frac{1}{2}$, while the Cesàro sum of the sequence $1,1,0,1,1,0,\mathrm{\dots}$ is $\frac{1}{3}$.
Lemma 5.
If $f\mathrm{:}A\mathrm{\to}\mathrm{R}$ is summable, then for any set $C$ that contains $A$, the function $g\mathrm{:}C\mathrm{\to}\mathrm{R}$ defined by
$$g(c)=\{\begin{array}{cc}f(c)\hfill & c\in A\hfill \\ 0\hfill & \text{\mathit{o}\mathit{t}\u210e\mathit{e}\mathit{r}\mathit{w}\mathit{i}\mathit{s}\mathit{e}}\hfill \end{array}$$ 
is summable, and
$$\sum _{a\in A}f(a)=\sum _{c\in C}g(c).$$ 
Proof.
Let $\sigma ={\sum}_{a\in A}f(a)$. For $\u03f5>0$, there is some ${F}_{\u03f5}\in {\mathcal{P}}_{0}(A)$ such that ${F}_{\u03f5}\subseteq F\in {\mathcal{P}}_{0}(A)$ implies that $$. If ${F}_{\u03f5}\subseteq H\in {\mathcal{P}}_{0}(C)$, then, as ${F}_{\u03f5}\subseteq H\cap A\in {\mathcal{P}}_{0}(A)$,
${S}_{g}(H)\sigma $  $=$  $\left{\displaystyle \sum _{c\in H}}g(c)\sigma \right$  
$=$  $\left{\displaystyle \sum _{c\in H\cap A}}g(c)+{\displaystyle \sum _{c\in H\setminus A}}g(c)\sigma \right$  
$=$  $\left{\displaystyle \sum _{a\in H\cap A}}f(a)+{\displaystyle \sum _{c\in H\setminus A}}0\sigma \right$  
$=$  ${S}_{f}(H\cap A)\sigma $  
$$  $\u03f5.$ 
This shows that ${S}_{g}$ converges to $\sigma $. ∎
The previous two lemmas are useful, and also convince us that unordered summation works similarly to finite sums. We now establish conditions under which a function is summable.
Lemma 6.
If $f\mathrm{:}A\mathrm{\to}\mathrm{R}$ is nonnegative and there is some $M\mathrm{\in}\mathrm{R}$ such that $F\mathrm{\in}{\mathrm{P}}_{\mathrm{0}}\mathit{}\mathrm{(}A\mathrm{)}$ for all ${S}_{f}\mathit{}\mathrm{(}F\mathrm{)}\mathrm{\le}M$, then $f$ is summable. If $f\mathrm{:}A\mathrm{\to}\mathrm{R}$ is nonnegative and summable, then ${S}_{f}\mathit{}\mathrm{(}F\mathrm{)}\mathrm{\le}{\mathrm{\sum}}_{a\mathrm{\in}A}f\mathit{}\mathrm{(}a\mathrm{)}$ for all $F\mathrm{\in}{\mathrm{P}}_{\mathrm{0}}\mathit{}\mathrm{(}A\mathrm{)}$.
Proof.
Suppose there is some $M\in \mathbb{R}$ such that if $F\in {\mathcal{P}}_{0}(A)$ then ${S}_{f}(F)\le M$. That is, $M$ is an upper bound for the range of ${S}_{f}$. Because $f$ is nonnegative, the net ${S}_{f}$ is increasing. We apply Lemma 3, which tells us that ${S}_{f}$ converges to the supremum of its range. That ${S}_{f}$ converges means that $f$ is summable.
Suppose that $f$ is summable, and let $\sigma ={\sum}_{a\in A}f(a)$. Suppose by contradiction that there is some ${F}_{0}\in {\mathcal{P}}_{0}(A)$ such that ${S}_{f}({F}_{0})>\sigma $, and let $\u03f5={S}_{f}({F}_{0})\sigma $. Then there is some ${F}_{\u03f5}\in {\mathcal{P}}_{0}(A)$ such that ${F}_{\u03f5}\subseteq F\in {\mathcal{P}}_{0}(A)$ implies that $$. As ${F}_{\u03f5}\subseteq {F}_{0}\cup {F}_{\u03f5}\in {\mathcal{P}}_{0}(A)$, we have $$, and hence
$$ 
But ${F}_{0}$ is contained in ${F}_{0}\cup {F}_{\u03f5}$ and $f$ is nonnegative, so
$${S}_{f}({F}_{0})\le {S}_{f}({F}_{0}\cup {F}_{\u03f5}),$$ 
which gives $$, a contradiction. Therefore, there is no $F\in {\mathcal{P}}_{0}(A)$ for which ${S}_{f}({F}_{0})>\sigma $. ∎
Lemma 7.
Suppose that $f\mathrm{:}A\mathrm{\to}\mathrm{R}$ is a function and that ${A}_{\mathrm{+}}\mathrm{=}\mathrm{\{}a\mathrm{\in}A\mathrm{:}f\mathit{}\mathrm{(}a\mathrm{)}\mathrm{\ge}\mathrm{0}\mathrm{\}}$ and ${A}_{\mathrm{}}\mathrm{=}\mathrm{\{}a\mathrm{\in}A\mathrm{:}f\mathit{}\mathrm{(}a\mathrm{)}\mathrm{\le}\mathrm{0}\mathrm{\}}$. Then, $f$ is summable if and only if $f$ is summable over both ${A}_{\mathrm{+}}$ and ${A}_{\mathrm{}}$. If $f$ is summable, then
$$\sum _{a\in A}f(a)=\sum _{a\in {A}_{+}}f(a)+\sum _{a\in {A}_{}}f(a).$$ 
Proof.
Suppose that $f$ is summable. Because $f$ is summable, there is some $E\in {\mathcal{P}}_{0}(A)$ such that $E\subseteq F\in {\mathcal{P}}_{0}(A)$ implies that $$. Define
$${E}_{+}=\{a\in E:f(a)\ge 0\}\in {\mathcal{P}}_{0}({A}_{+}),{E}_{}=\{a\in E:f(a)\le 0\}\in {\mathcal{P}}_{0}({A}_{}).$$ 
If $G\in {\mathcal{P}}_{0}({A}_{+})$ then $E\subseteq G\cup E\in {\mathcal{P}}_{0}(A)$, and hence $$. We have
$${S}_{{f}_{+}}(G)=\sum _{a\in G}f(a)\le \sum _{a\in G\cup {E}_{+}}f(a)=\sum _{a\in G\cup E}f(a)\sum _{a\in {E}_{}}f(a),$$ 
and hence
$$ 
That is, $\sigma +1{S}_{f}({E}_{})$ is an upper bound for the range of ${S}_{{f}_{+}}$. The net ${S}_{{f}_{+}}$ is increasing, hence applying Lemma 3 we get that ${S}_{{f}_{+}}$ converges. That is, ${f}_{+}$ is summable. If $H\in {\mathcal{P}}_{0}({A}_{})$, then $E\subseteq H\cup E\in {\mathcal{P}}_{0}(A)$, and hence $$. We have
$${S}_{{f}_{}}(H)=\sum _{a\in H}f(a)\ge \sum _{a\in H\cup {E}_{}}f(a)=\sum _{a\in H\cup E}f(a)\sum _{a\in {E}_{+}}f(a),$$ 
and then
$${S}_{{f}_{}}(H)\ge {S}_{f}(H\cup E){S}_{f}({E}_{+})>\sigma 1{S}_{f}({E}_{+}),$$ 
showing that $\sigma +1+{S}_{f}({E}_{+})$ is an upper bound for the net ${S}_{{f}_{}}$. As ${S}_{{f}_{}}$ is increasing, by Lemma 3 it converges, and it follows that ${S}_{{f}_{}}$ converges. That is, ${f}_{}$ is summable.
Suppose that $f$ is summable over both ${A}_{+}$ and ${A}_{}$. Let ${f}_{+}$ be the restriction of $f$ to ${A}_{+}$ and let ${f}_{+}$ be the restriction of $f$ to ${A}_{+}$, and define ${g}_{+},{g}_{}:A\to \mathbb{R}$ by
$${g}_{+}(a)=\{\begin{array}{cc}f(a)\hfill & a\in {A}_{+}\hfill \\ 0\hfill & a\in {A}_{},\hfill \end{array}\mathit{\hspace{1em}\hspace{1em}}{g}_{}(a)=\{\begin{array}{cc}0\hfill & a\in {A}_{+}\hfill \\ f(a)\hfill & a\in {A}_{}.\hfill \end{array}$$ 
By Lemma 5, ${f}_{+}$ being summable implies that ${g}_{+}$ is summable, with
$$\sum _{a\in {A}_{+}}{f}_{+}(a)=\sum _{a\in A}{g}_{+}(a),$$ 
and ${f}_{}$ being summable implies that ${g}_{}$ is summable, with
$$\sum _{a\in {A}_{}}{f}_{}(a)=\sum _{a\in A}{g}_{}(a).$$ 
But $f={g}_{+}+{g}_{}$, so by Lemma 4 we get that $f$ is summable, with
$$\sum _{a\in A}f(a)=\sum _{a\in A}{g}_{+}(a)+\sum _{a\in A}{g}_{}(a)=\sum _{a\in {A}_{+}}{f}_{+}(a)+\sum _{a\in {A}_{}}{f}_{}(a).$$ 
∎
If $f:A\to \mathbb{R}$ is a function, we define $f:A\to \mathbb{R}$ by $f(a)=f(a)$.
Theorem 8.
If $f\mathrm{:}A\mathrm{\to}\mathrm{R}$ is a function, then $f$ is summable if and only if $\mathrm{}f\mathrm{}$ is summable.
Proof.
Let ${A}_{+}=\{a\in A:f(a)\ge 0\}$ and ${A}_{}=\{a\in A:f(a)\le 0\}$, and let ${f}_{+}$ and ${f}_{}$ be the restrictions of $f$ to ${A}_{+}$ and ${A}_{}$ respectively. Suppose that $f$ is summable. Then by Lemma 7 we get that ${f}_{+}$ is summable and ${f}_{}$ is summable. Let $F\in {\mathcal{P}}_{0}(A)$ and write ${F}_{+}=\{a\in F:f(a)\ge 0\}$, ${F}_{}=\{a\in F:f(a)\le 0\}$. We have
$${S}_{f}(F)=\sum _{a\in F}f(a)=\sum _{a\in {F}_{+}}f(a)\sum _{a\in {F}_{}}f(a)={S}_{{f}_{+}}({F}_{+}){S}_{{f}_{}}({F}_{}).$$ 
But by Lemma 6, because the net ${S}_{{f}_{+}}$ is increasing we have ${S}_{{f}_{+}}({F}_{+})\le {\sum}_{a\in {A}_{+}}{f}_{+}(a)$, and because the net ${S}_{{f}_{}}$ is increasing we have ${S}_{{f}_{}}({F}_{})\le {\sum}_{a\in {A}_{}}{f}_{}(a)$. Therefore, ${\sum}_{a\in {A}_{+}}{f}_{+}(a){\sum}_{a\in {A}_{}}{f}_{}(a)$ is an upper bound for the range of ${S}_{f}$. Moreover, ${S}_{f}$ is increasing, so by Lemma 6 it follows that ${S}_{f}$ converges, i.e. that $f$ is summable.
Suppose that $f$ is summable. By Lemma 6, for any $F\in {\mathcal{P}}_{0}({A}_{+})$ we have
$${S}_{{f}_{+}}(F)={S}_{f}(F)\le \sum _{a\in A}f(a),$$ 
i.e., ${\sum}_{a\in A}f(a)$ is an upper bound for the range of ${S}_{{f}_{+}}$. As ${S}_{{f}_{+}}$ is increasing, by Lemma 6 it follows that ${S}_{{f}_{+}}$ converges, i.e., that ${f}_{+}$ is summable. Because ${S}_{{f}_{}}$ is increasing, we likewise get that ${S}_{{f}_{}}$ converges and hence that ${S}_{{f}_{}}$ converges, i.e. that ${f}_{}$ is summable. Now applying Lemma 7, we get that $f$ is summable. ∎
Theorem 9.
If $f\mathrm{:}A\mathrm{\to}\mathrm{R}$ is summable, then $\mathrm{\{}a\mathrm{\in}A\mathrm{:}f\mathit{}\mathrm{(}a\mathrm{)}\mathrm{\ne}\mathrm{0}\mathrm{\}}$ is countable.
Proof.
Suppose by contradiction that $\{a\in A:f(a)\ne 0\}$ is uncountable. We have
$$\{a\in A:f(a)\ne 0\}=\{a\in A:f(a)>0\}=\bigcup _{n\in \mathbb{N}}\{a\in A:f(a)\ge \frac{1}{n}\}.$$ 
Since this is a countable union, there is some $n\in \mathbb{N}$ such that $\{a\in A:f(a)\ge \frac{1}{n}\}$ is uncountable; in particular, this set is infinite. Because $f$ is summable, by Theorem 8 we have that $f$ is summable, with unordered sum $\sigma $. Hence, there is some ${F}_{1}\in {\mathcal{P}}_{0}(A)$ such that ${F}_{1}\subseteq F\in {\mathcal{P}}_{0}(A)$ implies that $$. Let $F$ be a finite subset of $\{a\in A:f(a)\ge \frac{1}{n}\}$ with at least $n(\sigma +1)$ elements. Then
$${S}_{f}(F\cup {F}_{1})=\sum _{a\in F\cup {F}_{1}}f(a)\ge \sum _{a\in F}f(a)\ge n(\sigma +1)\cdot \frac{1}{n}=\sigma +1.$$ 
But ${F}_{1}\subseteq F\cup {F}_{1}\in {\mathcal{P}}_{0}(A)$, so $$, a contradiction. Therefore, $\{a\in A:f(a)\ne 0\}$ is countable. ∎
8 References
McArthur [52]
Schaefer [64, p. 120]
Roytvarf [63, p. 282]
McShane [53]
Diestel, Jarchow and Tonge [13]
Remmert [59, p. 29]
Sorenson [67]
Lattice sums [18]
Kadets and Kadets [40]
Manning [50]
Bottazini [2]
Boyer [4]
Weil [74]
Smithies [66]
Dugac [16]
Whiteside [75]
Schaefer [65]
Cauchy [5]
Polya [57]
Lakatos [44]
Krantz [43]
Cunha [12]
Youschkevitch [77]
Bromwich [6, p. 74, Art. 28]
Tucciarone [69]
Fraser [27]
Cowen [11]
Spence [68]
Jahnke [38]
Epple [21]
Mascré [51]
Rosenthal [61]
Freniche [28]
Goursat [29, p. 348]
9 Probability
Baker [1]
Nathan [55]
Nover and Harris [56]
Colyvan [10]
Liouville [49, pp. 74–75]
Chrystal [9, p. 118]
Jordan [39, p. 277, Theorem 291]
Cayley [8, a]
Harkness and Morley [35, p. 66]
Hofmann [37]
Ferreirós [26]
Brouncker [7]
Roy [62]
Wallis [72]
Bourbaki [3, p. 261, chapter III, §5.1]
Pringsheim [58]
Dutka [17]
Grünbaum [34]
Hinton and Martin [36]
Gersonides [42]
Watling [73]
Moore [54]
Wojtaszczyk [76, Chapter 7]
References
 [1] (2007) Putting expectations in order. Philosophy of Science 74 (5), pp. 692–700. Cited by: §9.
 [2] (1986) The higher calculus: a history of real and complex analysis from Euler to Weierstrass. Springer. Note: Translated from the Italian by Warren Van Egmond Cited by: §8.
 [3] (1989) General topology, chapters 1–4. Elements of Mathematics, Springer. Cited by: §9.
 [4] (1959) The history of the calculus and its conceptual development. Dover Publications, New York. Cited by: §8.
 [5] (2009) Cauchy’s Cours d’analyse: an annotated translation. Sources and Studies in the History of Mathematical and Physical Sciences, Springer. Cited by: §8.
 [6] (1931) An introduction to the theory of infinite series. second edition, Macmillan and Co., London. Cited by: §8.
 [7] (1668) The squaring of the hyperbola, by an infinite series of rational numbers, together with its demonstration, by that eminent mathematician, the right honourable the Lord Viscount Brouncker. Phil. Trans. 3, pp. 645–649. Cited by: §9.
 [8] (1886) Series. In Encyclopædia Britannica, vol. XXI, pp. 677–682. Note: Collected Mathematical Papers, vol. XI, pp. 617–627 Cited by: §9.
 [9] (1889) Algebra: an elementary textbook for the higher classes of secondary schools and for colleges, part II. Adam and Charles Black, Edinburgh. Cited by: §9.
 [10] (2006) No expectations. Mind 115 (459), pp. 695–702. Cited by: §9.
 [11] (1980) Rearranging the alternating harmonic series. Amer. Math. Monthly 87 (10), pp. 817–819. Cited by: §8.
 [12] (1988) Anastácio da Cunha and the concept of convergent series. Arch. Hist. Exact Sci. 39 (1), pp. 1–12. Cited by: §8.
 [13] (1995) Absolutely summing operators. Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press. Cited by: §8.
 [14] (1837) Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, pp. 45–81. Note: Werke, vol. I, pp. 313–342 Cited by: §3.
 [15] (1999) Lectures on number theory. History of Mathematics, Vol. 16, American Mathematical Society, Providence, RI. Note: Supplements by R. Dedekind, translated from the German by John Stilwell Cited by: §3.
 [16] (1973) Éléments d’analyse de Karl Weierstrass. Arch. History Exact Sci. 10 (1/2), pp. 41–176. Cited by: §8.
 [17] (1988) On the St. Petersburg paradox. Arch. Hist. Exact Sci. 39 (1), pp. 13–39. Cited by: §9.
 [18] (2010) Gittersummen der Festkörperphysik. Wie sinnvoll ist die Sprache des Unendlichen für die Dinge des alltäglichen Lebens?. Mitt. Math. Ges. Hamburg 29, pp. 71–96. Cited by: §8.
 [19] (1998) Annotationes historicae. Mitt. Dtsch. Math.Ver. (1), pp. 23–24. Cited by: §3.
 [20] (2005) The life and work of Gustav Lejeune Dirichlet (1805–1859). In Analytic Number Theory: A Tribute to Gauss and Dirichlet, W. Duke and Y. Tschinkel (Eds.), Clay Mathematics Proceedings, Vol. 7, pp. 1–38. Cited by: §3.
 [21] (2003) The end of the science of quantity: foundations of analysis, 1860–1910. In A History of Analysis, H. N. Jahnke (Ed.), History of Mathematics, Vol. 24, pp. 291–324. Cited by: §8.
 [22] (1755) Institutiones calculi differentialis, vol. I. Academia Scientiarum Imperialis Petropolitanae, St. Petersburg. Note: E212, Opera omnia I.10 Cited by: §2.
 [23] (1999) The first modern definition of the sum of a divergent series: an aspect of the rise of 20th century mathematics. Arch. Hist. Exact Sci. 54 (2), pp. 101–135. Cited by: §9.
 [24] (2000) True and fictitious quantities in Leibniz’s theory of series. Studia Leibnitiana 32 (1), pp. 43–67. Cited by: §9.
 [25] (2008) The rise and development of the theory of series up to the early 1820s. Sources and Studies in the History of Mathematics and Physical Sciences, Springer. Cited by: §9.
 [26] (2007) Labyrinth of thought: a history of set theory and its role in modern mathematics. second edition, Birkhäuser. Cited by: §9.
 [27] (1989) The calculus as algebraic analysis: some observations on mathematical analysis in the 18th century. Arch. Hist. Exact Sci. 39 (4), pp. 317–335. Cited by: §8.
 [28] (2010) On Riemann’s rearrangement theorem for the alternating harmonic series. Amer. Math. Monthly 117 (5), pp. 442–448. Cited by: §8.
 [29] (1904) A course in mathematical analysis, vol. I. Ginn and Company, Boston. Note: Translated from the French by Earle Raymond Hedrick Cited by: §8.
 [30] (1970) Bolzano, Cauchy and the “new analysis” of the early nineteenth century. Arch. Hist. Exact Sci. 6 (5), pp. 372–400. Cited by: §8.
 [31] (1970) The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. Cited by: §8.
 [32] (1980) The emergence of mathematical analysis and its foundational progress, 1780–1880. In From the Calculus to Set Theory, 1630–1910, I. GrattanGuinness (Ed.), pp. 94–148. Cited by: §8.
 [33] (1990) Thus it mysteriously appears: impressions of Laplace’s use of series. In Rechnen mit dem Unendlichen: Beiträge zur Entwicklung eines kontroversen Gegenstandes, D. D. Spalt (Ed.), pp. 95–102. Cited by: §8.
 [34] (1969) Can an infinitude of operations be performed in a finite time?. Br. J. Philos. Sci. 20 (3), pp. 203–218. Cited by: §9.
 [35] (1893) A treatise on the theory of functions. Macmillan, London. Cited by: §9.
 [36] (1954) Achilles and the tortoise. Analysis 14 (3), pp. 56–68. Cited by: §9.
 [37] (1939) On the discovery of the logarithmic series and its development in England up to Cotes. National Mathematics Magazine 14 (1), pp. 37–45. Cited by: §9.
 [38] (2003) Algebraic analysis in the 18th century. In A History of Analysis, H. N. Jahnke (Ed.), History of Mathematics, Vol. 24, pp. 105–136. Cited by: §8.
 [39] (1893) Cours d’analyse de l’École Polytechnique, tome premier. second edition, GauthierVillars et fils, Paris. Cited by: §9.
 [40] (1997) Series in Banach spaces: conditional and unconditional convergence. Operator Theory: Advances and Applications, Vol. 94, Birkhäuser. Note: Translated from the Russian by Andrei Iacob Cited by: §8.
 [41] (1955) General topology. University Series in Higher Mathematics, D. Van Nostrand Company. Cited by: §6.
 [42] (2006) Medieval infinities in mathematics and the contribution of Gersonides. History of Philosophy Quarterly 23 (2), pp. 95–116. Cited by: §9.
 [43] (2004) Creating more convergent series. Amer. Math. Monthly 111 (1), pp. 32–38. Cited by: §8.
 [44] (1976) Proofs and refutations. Cambridge University Press. Cited by: §8.
 [45] (1951) Differential and integral calculus. Chelsea Publishing Company, New York. Note: Translated from the German by Melvin Hausner and Martin Davis Cited by: §3, §4.
 [46] (1989) Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39 (3), pp. 195–245. Cited by: §8.
 [47] (2000) Controversies about numbers and functions. In The Growth of Mathematical Knowledge, E. Grosholz and H. Breger (Eds.), Sythese Library, Vol. 289, pp. 177–198. Cited by: §8.
 [48] (2008) Bernhard Riemann 1826–1866: turning points in the conception of mathematics. Birkhäuser. Note: Translated from the German by Abe Shenitzer Cited by: §8.
 [49] (1990) Joseph Liouville 1809–1882: master of pure and applied mathematics. Studies in the History of Mathematics and Physical Sciences, Vol. 15, Springer. Cited by: §9.
 [50] (1975) The emergence of the Weierstrassian approach to complex analysis. Arch. Hist. Exact Sci. 14 (4), pp. 297–383. Cited by: §8.
 [51] (2005) 1867 Bernhard Riemann, posthumous thesis on the representation of functions by trigonometric series. In Landmark Writings in Western Mathematics 1640–1940, I. GrattanGuinness (Ed.), pp. 491–505. Cited by: §8.
 [52] (1968) Series with sums invariant under rearrangement. Amer. Math. Monthly 75, pp. 729–731. Cited by: §8.
 [53] (1952) Partial orderings and MooreSmith limits. Amer. Math. Monthly 59 (1), pp. 1–11. Cited by: §8.
 [54] (1932) Summability of series. Amer. Math. Monthly 39 (2), pp. 62–71. Cited by: §9.
 [55] (1984) False expectations. Philosophy of Science 51 (1), pp. 128–136. Cited by: §9.
 [56] (2004) Vexing expectations. Mind 113 (450), pp. 237–249. Cited by: §9.
 [57] (1972) Problems and theorems in analysis, volume I. Die Grundlehren der mathematischen Wissenschaften, Vol. 193, Springer. Note: Translated from the German by D. Aeppli Cited by: §8.
 [58] (1898–1904) Irrationalzahlen und Konvergenz unendlicher Prozesse. In Enzyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band I, 1. Teil, W. F. Meyer (Ed.), pp. 47–146. Cited by: §9.
 [59] (1991) Theory of complex functions. Graduate Texts in Mathematics, Vol. 122, Springer. Note: Translated from the German by Robert B. Burckel Cited by: §8.
 [60] (1866/1867) Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13, pp. 87–132. Note: Gesammelte mathematische Werke, second ed., vol. II, 227–271 Cited by: §4.
 [61] (1987) The remarkable theorem of Lévy and Steinitz. Amer. Math. Monthly 94 (4), pp. 342–351. Cited by: §8.
 [62] (2011) Sources in the development of mathematics. Cambridge University Press. Note: Infinite series and products from the fifteenth to the twentyfirst century Cited by: §9.
 [63] (2013) Thinking in problems: how mathematicians find creative solutions. Birkhäuser. Cited by: §8.
 [64] (1999) Topological vector spaces. second edition, Graduate Texts in Mathematics, Vol. 3, Springer. Note: With the assistance of M. P. Wolff Cited by: §8.
 [65] (1986) Sums of rearranged series. College Math. J. 17 (1), pp. 66–70. Cited by: §8.
 [66] (1986) Cauchy’s conception of rigour in analysis. Arch. Hist. Exact Sci. 36 (1), pp. 41–61. Cited by: §8.
 [67] (2010) Throwing some light on the vast darkness that is analysis: Niels Henrik Abel’s critical revision and the concept of absolute convergence. Centaurus 52 (1), pp. 38–72. Cited by: §8.
 [68] (1987) How many elements are in a union of sets?. The Mathematics Teacher 80 (8), pp. 666–670, 681. Cited by: §8.
 [69] (1973) The development of the theory of summable divergent series from 1880 to 1925. Arch. Hist. Exact Sci. 10 (1/2), pp. 1–40. Cited by: §8.
 [70] (1992) James Stirling’s early work on acceleration of convergence. Arch. Hist. Exact Sci. 45 (2), pp. 105–125. Cited by: §8.
 [71] (2003) James Stirling’s Methodus differentialis: an annotated translation of Stirling’s text. Sources and Studies in the History of Mathematics and Physical Sciences, Springer. Cited by: §8.
 [72] (2004) The arithmetic of infinitesimals. Sources and Studies in the History of Mathematics and Physical Sciences, Springer. Note: Translated from the Latin and with an introduction by Jaequeline A. Stedall Cited by: §9.
 [73] (1952) The sum of an infinite series. Analysis 13 (2), pp. 39–46. Cited by: §9.
 [74] (1976) Elliptic functions according to Eisenstein and Kronecker. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 88, Springer. Cited by: §8.
 [75] (1961) Patterns of mathematical thought in the later seventeenth century. Arch. Hist. Exact Sci. 1 (3), pp. 179–388. Cited by: §8.
 [76] (1997) A mathematical introduction to wavelets. London Mathematical Society Student Texts, Vol. 37, Cambridge University Press. Cited by: §9.
 [77] (1976) The concept of function up to the middle of the 19th century. Arch. Hist. Exact Sci. 16 (1), pp. 37–85. Cited by: §8.