§3. Bolzano-Weierstrass and Cauchy's criterion
243
The notation of Theorem 17 was chosen to make manifest its analogy with
the theorems on "passing to the limit under the J sign" or on "dominated
convergence" in the theory of integration. These say, roughly, that if one
has a function f(x, t) defined on I x T, where I is an arbitrary interval and
T a subset of IR or C, if the function x ........ f(x, t) is absolutely integrable
on I for any t E T, if it converges simply to a limit function f(x) when
t E T tends to a limit a and if there exists an integrable function g( x) with
positive values satisfying
If(x, t)1 :::; g(x) for all x and t,
then
lim r f(x, t)dx = r f(x)dx.
t-+a if
if
This is the analogue of normal convergence when one replaces series, which
are sums of terms depending on a "discrete" variable, by integrals, which
are (or claim to be, chez Leibniz) sums of terms depending on a "continuous" variable: the variable of integration x plays the role of the index of
summation n, the integral taken over I replaces the sum taken over N, the
concept of an "absolutely integrable" function replaces that of an absolutely convergent series, and finally the existence of an integrable function
g(x) which "dominates" all the f(x, t) plays the role of normal convergence.