540 
DERIVATES, EXTREMES, VARIATION 
Hence a fortiori, | | < 
Now the sum 
i=vi 
(9 
converges if /x > 0. Hence _H Pj 3 and H p may be made as small as 
we choose, by taking p sufficiently large. Let us note that by 91, 
1 1 
ir P <- 
np* 
Thus if /x = Min (a, /3), 
s M a 
B < I D„ I + | | < 2 X 2 JfA, 
for a sufficiently large r. 
TFe consider finally C. We have 
0 < I AI + I I 
e 
< 3’ 
^ rrr 2 I + A) | + j-^-r 2 a n | ^(e TO „)| + | G s 
I ” | 4+1 I'M i+1 
< O x + Q % 4- C 3 . 
From 9) we see that 
(7 3 < M X H S < 
for s sufficiently large. Since g(x) is continuous in $8, 
\g(*)\<N. 
Hence * n T ht i 
C, and 0 2 < d-r 2) < — - 
1 2_ U .Ti»s + - + »- ¡All-*-« 
< 
4+1 
w 
+ a 4- /3 s 1+a+ 0 
1 + « -|- /3 s a+ii ’ 
if s on using 10). 
\h\ 
Taking s still larger if necessary, we can make 
A. ^2 < 
(10 
*<!• 
0 
is 
t( 
ir 
SI 
Thus