Convergent Sequence of Continuous Real Functions is Integrable Termwise

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Theorem

Let $I = \closedint a b \subseteq \R$ be a closed real interval.

Let $\struct {\map CI, \norm {\, \cdot \,}_\infty}$ be the normed vector space of real-valued functions continuous on $I$ equipped with the supremum norm.

Let $\sequence {f_k}_{k \mathop \in \N_{>0} } \in C \closedint a b$ be a sequence.

Suppose $\ds \sum_{k \mathop = 1}^\infty f_k$ converges to $f$ in $\struct {\map C I, \norm {\, \cdot \,}_\infty}$.


Then:

$\ds \sum_{k \mathop = 1}^\infty \int_a^b f_k \rd t = \int_a^b \map f t \rd t$


Proof

\(\ds \int_a^b \map f t \rd t\) \(=\) \(\ds T f\) Definition of Riemann Integral Operator
\(\ds \) \(=\) \(\ds T \lim_{n \mathop \to \infty} s_n\) Definition of Convergent Series in Normed Vector Space
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} T s_n\) Riemann Integral Operator is Continuous Linear Transformation, Continuous Mappings preserve Convergent Sequences
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \int_a^b \map {s_n} t \rd t\) Definition of Riemann Integral Operator
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \sum_{k \mathop = 1}^n \int_a^b \map {f_k} t \rd t\) Definition of Sequence of Partial Sums
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^\infty \int_a^b \map {f_k} t \rd t\)

$\blacksquare$


Sources

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