An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 142.

The general problem of integration and its conditions.

241

jp=m
then we can separate the sum dp'Dp' into parts as follows:
P=1.= 2
'dp'Dp' + d'~i+D'2~I1 + 'Dp + d'/LlD'e+l + -
p=l 2 —1 -2
isolating the terms that refer to intervals that contain the dividing
points of the first partition; they are n - 1 in number. But since
D, denotes the greatest fluctuation in the whole interval d1, we have:
2 = 1 p =-R -- 2
Further, the sum of all the isolated terms is certainly smaller than
n - 1 times the product of the greatest d' of their intervals by the
greatest fluctuation D' occurring among them. Therefore:
p =nm p-=-n
dp'Dp,' < dDc + (n - 1l)'D'.
p-=1 p= 1
Now since we can arbitrarily diminish the values d', we can always
choose them so small that the product (n - l)d'D)'= o shall become
arbitrarily small; therefore we have:
p=m
dp'Dp' < + c
p=l
i. e. this sum also becomes arbitrarily small by suitable choice of m.
We now proceed with the proof of the above theorem as follows:
Let the entire interval from a to b be divided, in succession,
first into n1, then into n2, n3,.. n..,,. parts; and let:
each segment of the first partition:,(1), d ),... d,(1) be smaller than (,,
each segment of the second partition: d1(2), d2(2),...d,2( be smaller than ',,
each segment of the third partition: d(3), d2),...d,n3 be smaller than 8,
each segment of the Vth partition: cld(), d2(),... dn V(") be smaller than,,
let 61, 6 3, 3..,... form a series of positive numbers converging
to zero; and let the dividing points of the second partition include
all the dividing points of the first, and likewise let those of each
further partition include all the dividing points of the preceding one,
so that each interval d1, d.. is divided into new subdivisions.
Let G,(v) denote the superior limit of the function f(x) within the
interval d{(v), taking the sign into account, and similarly g,^(V) its
inferior limit. Then let us form the suns:
HIARNACIK Calculus. 16