FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 123 



As the left-hand member is not negative, it follows that, for all values of m, 



X - 1 



and hence that the series 



t* convergent. 



6. Since the n th term of a convergent series tends to zero, as n increases indefi- 

 nitely, it follows from the result obtained in the preceding paragraph that 



lim |V(t)/(*)<fc~0 .......... (9) 



n * oc J a 



Recalling the hypothesis in regard to f(s), it is clear that this relation is true for 

 all limited functions which have a Lebesgue integral in (a, 6). When it is possible 

 to find a number \\i which, for all values of n, is greater than the upper limit of 

 | i/ B (s) | in (a, b), we may show that (9) is also true for all unlimited functions which 

 possess a Lebesgue integral in (a, 6). For, assuming that f(s) is unlimited, let us 

 select a positive number N, and define a function/, (s) in (a, b) by the rule 



/.(*) =/() 



= |/(.)|>N. 



If e is any assigned positive number, it is known that N may be chosen great 

 enough to ensure that 



-/()!*< 



Hence, with this choice of N, we have 



| J a (i 



Again, since / (s) is limited, we can find a positive integer such that, for this and 

 all greater values of n, the numerical value of 



is less than . For these values of n we therefore have 



* This is sometimes called BESSEL'S inequality, vide the footnote on p. 66 of BOCHKR'S tract, ' An 

 Introduction to the Study of Integral Equations' (1909). 



R 2 



