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.

~ 103.

Uniqueness of the limiting value.

175

f(xk-l) another value of f that belongs to a point within
the interval dk.
In like manner, S" arises from S' by the occurrence in place of
the term d'kf(x'k-l) in S' of another value of f that belongs to a
point in the interval d'k; where d'k denotes that one of the n' partial
intervals that begins at the point x'k-1; and so on.
But now since the function f is continuous, at each point a
finite interval can be discovered wherein the various values of f differ
by less than an arbitrarily small finite quantity e. Therefore by continued subdivision the intervals can certainly be made so small, that
in each of them the absolute differences of the various values of f
shall be smaller than e; let the number of these intervals be n(k),
the respective sum S(k), when we advance to any of the further partitions
we have:
abs [S) - S(k+t)] < E{d1 + d2 +. Cn(k) };
but as the total interval is always equal to (x - a), this difference
is smaller than E(x - a).
When therefore the partition has advanced so far, that in each
interval the fluctuations of f are smaller than -, any further
G - a
partitions can alter the amount of S(k) only by less than 6; therefore
the series S, S', S", etc., approximates to a determinate limiting value.
But it must still be investigated, whether this limiting value
depends on the original partition into n intervals and the consequent partition of each of these into smaller intervals, or whether
it is quite indifferent in what way the total interval from a to x is
broken up into subdivisions that ultimately decrease below any finite
amount.*) That the latter is the case, appears from the following
consideration.
Let the original partition be into m parts, the corresponding sum
being S,. By further dividing these intervals we obtain as before a
series of values S,(1), S(),..... S..., the numbers of intervals being
n', n"... m) etc.. Let the partition have advanced so far, that
each further partition can alter the value of S1(k) only by less than 6.
Now let us conceive these two partitions: -into me(), and into n(k)
intervals, combined into a single one, then to it belongs a sum 2. that
differs from S(k) as well as from Si(k) by a quantity smaller than 6;
for, this third partition arising from their combination is to be
regarded as a continuation of each of the two former. Abs [S(k) - S1(k)]
is therefore smaller than the arbitrarily small quantity 26, i. e. the
series S1 has the same limiting value as the series S.

') This is investigated with still more detail in ~ 142.