TAOCP 1.2.3 Exercise 6
Let A=\sum_{R(j)} a_j,\qquad B=\sum_{S(j)} a_j,\qquad C=\sum_{R(j)\text{ or }S(j)} a_j,\qquad D=\sum_{R(j)\text{ and }S(j)} a_j .
Section 1.2.3: Sums and Products
Exercise 6. [HM20] Prove that rule (d) is valid for an arbitrary infinite series, provided that any three of the four sums exist.
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Solution
Let
$$ A=\sum_{R(j)} a_j,\qquad B=\sum_{S(j)} a_j,\qquad C=\sum_{R(j)\text{ or }S(j)} a_j,\qquad D=\sum_{R(j)\text{ and }S(j)} a_j . $$
We must prove that
$$ A+B=C+D, $$
provided that any three of the four sums exist.
Using Iverson's convention,
$$ [R(j)]+[S(j)] = [R(j)\text{ or }S(j)] + [R(j)\text{ and }S(j)] $$
for every integer $j$. Indeed, this is verified by the four possible truth values of $R(j)$ and $S(j)$.
Now fix $N\ge0$, and define the finite partial sums
$$ A_N=\sum_{|j|\le N} a_j [R(j)], \qquad B_N=\sum_{|j|\le N} a_j [S(j)], $$
$$ C_N=\sum_{|j|\le N} a_j [R(j)\text{ or }S(j)], \qquad D_N=\sum_{|j|\le N} a_j [R(j)\text{ and }S(j)]. $$
Since these are finite sums, we may sum the pointwise identity over all $j$ with $|j|\le N$. Thus
$$ A_N+B_N=C_N+D_N $$
for every $N$.
Assume now that any three of the four infinite sums $A,B,C,D$ exist. We show that the fourth also exists and that the identity holds.
Without loss of generality, suppose $A,B,C$ exist. Then
$$ \lim_{N\to\infty}A_N=A,\qquad \lim_{N\to\infty}B_N=B,\qquad \lim_{N\to\infty}C_N=C. $$
From the finite identity,
$$ D_N=A_N+B_N-C_N. $$
Taking limits and using the ordinary limit laws,
$$ \lim_{N\to\infty}D_N = A+B-C. $$
Hence $D$ exists and
$$ D=A+B-C. $$
Therefore
$$ A+B=C+D. $$
This proves the desired identity when $A,B,C$ exist.
The same argument works if any other three of the four sums exist: from
$$ A_N+B_N=C_N+D_N $$
one solves for the missing partial sum and passes to the limit. Thus the existence of any three sums implies the existence of the fourth, and in every case
$$ \sum_{R(j)} a_j+\sum_{S(j)} a_j = \sum_{R(j)\text{ or }S(j)} a_j + \sum_{R(j)\text{ and }S(j)} a_j . $$
Hence rule (d) is valid for an arbitrary infinite series, provided that any three of the four sums exist. ∎