Corollary 1

Tag 088Z, permanent. Cite it as [Bourbaki, Tag 088Z].

Functions of a Real Variable, chapter II, § 3. DERIVATIVES AND INTEGRALS OF FUNCTIONS DEPENDING ON A PARAMETER

Let $(f_n)$ be a sequence of regulated functions on a compact interval $I = [a, b]$ . If the sequence $(f_n)$ converges uniformly on I to a (regulated) function f, one has

\lim_{n \to \infty} \int_a^b f_n(t) \, dt = \int_a^b f(t) \, dt.

In particular, if a series whose general term $u_n$ is a regulated function on I, converges uniformly to f on I, then the series with general term $\int_a^b u_n(t) \, dt$ is convergent and its sum is $\int_a^b f(t) \, dt$ ("term-by-term integration of a uniformly convergent series").

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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