894 

 has its maximum modulus. From this it follows 



and consequently 



1 1 



/m [a,„(a'„ -f «')]'«< /im |a,„(ar,„)|'». . . . (52) 



The left-hand member is simply equal to a («'), the upper limit for 



the domain («') of 



1 



(ix = Urn \a,n{a;)\ '« , ■ 



and this because all functions a„, have their maximum modulus 

 exactly in the point jc^ -{- a of the domain («'). For the right-hand 

 member it would not hold without more that it is equal to a{a), 

 if we did not adhere to the so-called uniformity-supposition of N'. 4, 

 that corresponding to any given arbitrarily small number e, there 

 is for all points .v of the domain («) one integer du , such that 



1 

 |«»« («) I »" < aa: + f , for m > w, . 

 On this supposition it is not possible that the right-hand member 

 of (52) differs from «(«). For, let us suppose that this were the 

 case, and that we had for example 



1 

 lim \a,n {xm)\ "' = a (a) -\- d, (53) 



where d" is a certain positive number. Take s =: ^d. We have then 

 for m > mc 



1 



km («m)|"'< a^^ f ^(f<a («) + è (f , 



contrary to (53). That the limit in the right-hand member of (52) 

 cannot be less than a{a), appears at once, if we observe that for 

 an arbitrary point x of the domain («) 



1 1 



|a;„(^m)|'«> |am(^)|'», 



consequently 



lim ja,„ (;c,„)j'«> % 

 and therefore also, as this inequality holds for any point x of («), 



1 



lim \a,nix,n)\ "'>a{a) 



m = 00 



The inequality (52) passes consequently into 



