Corollary 1

Tag 0894, permanent. Cite it as [Bourbaki, Tag 0894].

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 an arbitrary interval $I$ , converging uniformly to a function $f$ on each compact interval contained in $I$ ; if the integral $\int_I f_n(t)\,dt$ is uniformly convergent, then the integral $\int_I f(t)\,dt$ is convergent, and

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

(9)

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

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