1549901369-Elements_of_Real_Analysis__Denlinger_

532 Chapter 8 • Infinite Series of Real Numbers

We have just seen that all the row sums of both of these nonnegative

matrices (double series) converge. Further, 'r/n EN,

L n(oo) L atj ::::; L n(oo L laijl ) ::::; sum by rows of L oo laijl, and

i=l J=l i=l J=l i,J=l

Thus, since their partial sums are bounded, the "sum by rows" of each of
the double series (36) and (37) converge. Therefore, by Lemma 8.7.16, all their
column sums converge and their sums by columns equal their sums by rows.
That is,

L^00 (L^00 atj ) = L^00 (I:at^00 )

i=l j=l j=l i=l J

Therefore 'r/n E N,

= i~ c~

1

ati) -i~ c~

1

a;i). Taking the limit as n ---+ oo, we have

I:^00 Ri = I:^00 (I:^00 at ) -I:^00 (I:^00 a; )

i=l i=l j=l J i=l j=l J

= L^00 (L^00 atj ) - L^00 (L^00 a;j ).

j=l i=l j=l i=l

00

On the other hand, the sum by columns of L aij is

i,j=l

(38)

= j~l (~ aij) -j~l (~ a;j). Taking the limit as n ---+ oo, we have

I:^00 cj = I:^00 (I:^00 at ) -I:^00 (I:^00 a; ).
J=l.. J=l. i=l. J J. =l i. =l J (39)
00 00
Putting together (38) and (39), we have L Ri = L Cj.
i=l j=l