of an infinite series over an infinite range 7 



Hence 



rX' m' rX' m' 



I S Un {x)dx\^\ I S </> {x)f^ (x) I dx 



JX n=m+l J X n=m+l 



rX' m' 



< \<^{x)\t \fn {OC) I dx 



J X n = m 



^2g[ \d> {x) I dx. 



J X 



Now, given any positive number e, we can, since \<f) {x)\dx 



J a 

 converges, find Xo such that 



\(f){x)\dx< € 



J X 



forX, X'>Xo. Hence 



rX' to' 

 I 1 S ii„ (x) dx\< € 



Jx n = m+l 



for X, X' > Xf) and all rn, m . Thus the double limit 



exists. Further 



rx m 

 lim / S i<„ ix) dx 

 •■»,X-*"X J a n = \ 



n=l x=l 



and G^ 1 (ic) I is summable in C«, X) for all X greater than a. 

 Thus by (C 2) 



CX m rX 00 



lim X Un (x) dx, i u„ (x) dx 



J a n=l J a n=l 



exist and are equal and finite. 



All the conditions of (I) are now satisfied, and our theorem 

 follows. 



B. If we write r^ m 



rX m 

 Sm,X= I 2 \Un{x)\dx, 

 •I a n=l 



then lim 8m,x exists and is either finite or positive infinity. 



In the first case our theorem follows at once by (II). 

 In the second case, both the repeated limits 



lim (lim Sm,x), lim (lim 8m, x), 



m-^cD X^'x> X^oD m-*oo 



are infinite. Suppose now that 



S I I M„ (x) I dx 



»=1 J a 



