# Convergent Sequence of Continuous Real Functions is Integrable Termwise

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

$\blacksquare$