How to find that the alternating sum converges There's an exercise in Knuth's book, The art of computer programming sec 1.2.3, exercises first set, ex 8 that says:

Find an example of infinite series in which eq. (the following) is false:

$$
\sum_{R(i)}\sum_{S(j)} a_{ij} = \sum_{S(j)}\sum_{R(i)} a_{ij}
$$

An example is to take $a_{(i+1)i}=1$, $a_{i(i+1)}=-1$, others $a_{ij}=0$.
So this is a "sign sum", the left side starts with $+1$ and the right with $-1$. How do you prove that the left side is $-1$, and the right is $+1$? (The sum is infinite, starts at $i=0$, $j=0$)
The idea of the proof is that 
$$\sum_{j\in\mathbb{N}} a_{ij} = \left\{\begin{array}{cc} -1 &amp; i=1\\
0 &amp; i\not = 1 \end{array}\right.,
$$
and 
$$\sum_{i\in \mathbb{N}} a_{ij} = \left\{ \begin{array}{cc} 1&amp;j=1\\  0&amp; j\not=1\end{array}\right. $$

It would help to write down the values of $a_{ij}$ for $(i,j)\in \{1,2,3,4\}\times \{1,2,3,4\}$, say.
look in to this article. that should be the right tool:
https://en.wikipedia.org/wiki/Mathematical_induction