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.

240 The definite integral as the limiting value of a sum. Bk. Ich. ch. VI.
to formulate the problem precisely, has also supplied its solution.
We restrict ourselves at first to functions which do not become infinite
anywhere in the finite interval from a to b, so that all the values of
the function are included between a superior limit that may be
denoted by G and an inferior that may be denoted by g, these being
positive or negative.
The function must be one that is defined without exception for
this interval; i.e. its value belonging to each point is actually given.
The question proposed above may now be abridged into the words:
Under what hypotheses is szch a function integrable? The answer is:
If we denote the greatest fluctuation of the function, i. e. the
positive difference of its greatest and least values, in the interval from
a to x1 including those limits by D1, likewise between xi and x2 by
D2,... between x,,n and b by Dn, then the limit of the sum:
diDI + d2D2 + * + dn Dn (d, + d, + * * + d = b - a)
must be zero as the values of n increase, when simultaneously all the
quantities d converge to zero.
This is the necessary and sufficient condition. What takes place is:
When the above sum converges to zero for any law by which
the number of intervals increases arbitrarily, it always converges to
zero in whatever manner the quantities d are chosen and arbitrarily
diminished.
We prove this last statement in the first place as follows:
Suppose the number n already chosen so large, that the absolute
p=n
amount of dpDp, since its limiting value vanishes, is less than a.
p=1
Choosing another completely different subdivision into m parts, where
m > n, and the quantities d1', d2',... dn' are arbitrarily smaller
p=-m
than the least of the quantities d, we shall show that,cd'Dp'
p=1
also converges to zero. Let us consider the length a b simultaneously
divided into n intervals d and into m intervals d', then there will
be in each part d a certain number of the intervals d'; but, in general,
extremities of the intervals d' will not coincide with extremities of the
intervals d. Suppose:
a + dl + d2 + * * d2' < a + dI = xi < a + dl' + * * d'2.41~
a + dI'l + f2-' +.* ~ d' < XI + (2 = x2 < Ca + di' + * ~ * d', — i,
a + d + d' -- + ' e dC' < x2 + d3 = X3 < a + d,' + * * d'vl,. *.... * e.... o.. * o o.. o o o.