164 CHAPTER 6. INFINITE SERIES

interchange the order in which two limits are taken, you need a theorem which will allowyou to do it. Such theorems are often rather technical. One must never interchange limitsof any kind without agonizing over whether the symbol pushing is correct. In general,limits ruin algebra and also introduce things which are counter intuitive. Failure to keepthis in mind leads to mathematical disasters. Here is an example. This example is a littletechnical. It is placed here just to prove conclusively there is a question which needs to beconsidered.

Example 6.6.1 Consider the following picture which depicts some of the ordered pairs(m,n) where m,n are positive integers.

...0 0 c 0 −c

0 c 0 −c 0

b 0 −c 0 0

0 a 0 0 0

· · ·

The a,b,c are the values of amn. Thus ann = 0 for all n ≥ 1, a21 = a,a12 = b,am(m+1) =−cwhenever m > 1, and am(m−1) = c whenever m > 2. The numbers next to the point are thevalues of amn. You see ann = 0 for all n, a21 = a,a12 = b,amn = c for (m,n) on the liney = 1+ x whenever m > 1, and amn = −c for all (m,n) on the line y = x− 1 wheneverm > 2.

Then ∑∞m=1 amn = a if n = 1, ∑

∞m=1 amn = b− c if n = 2 and if n > 2,∑∞

m=1 amn = 0.Therefore, ∑

∞n=1 ∑

∞m=1 amn = a+b−c. Next observe that ∑

∞n=1 amn = b if m= 1,∑∞

n=1 amn =a+ c if m = 2, and ∑

∞n=1 amn = 0 if m > 2. Therefore, ∑

∞m=1 ∑

∞n=1 amn = b+ a+ c and so

the two sums are different. Moreover, you can see that by assigning different values of a,b,and c, you can get an example for any two different numbers desired.

Don’t become upset by this. It happens because, as indicated above, limits are takenin two different orders. An infinite sum always involves a limit and this illustrates whyyou must always remember this. This example in no way violates the commutative law ofaddition which has nothing to do with limits. However, it turns out that if ai j ≥ 0 for all i, j,then you can always interchange the order of summation. This is shown next and is basedon the following lemma. First, some notation should be discussed.

Definition 6.6.2 Let f (a,b)∈ [−∞,∞] for a∈A and b∈B where A,B are sets whichmeans that f (a,b) is either a number, ∞, or −∞. The symbol, +∞ is interpreted as a pointout at the end of the number line which is larger than every real number. Of course there isno such number. That is why it is called ∞. The symbol, −∞ is interpreted similarly. Thensupa∈A f (a,b) means sup(Sb) where Sb ≡ { f (a,b) : a ∈ A} .

Unlike limits, you can take the sup in different orders.

Lemma 6.6.3 Let f (a,b) ∈ [−∞,∞] for a ∈ A and b ∈ B where A,B are sets. Then

supa∈A

supb∈B

f (a,b) = supb∈B

supa∈A

f (a,b) .