Proof of term-by-term integration

Builds on Series of functions Linearity of the integral Uniform limits commute with integration Sum of a series

Proof

We apply the theorem on uniform limits and integration to the partial sums.

Let $a < b$ , and let each $f_n \colon [a, b] \to \R$ be Riemann integrable. Let $s_N = \sum_{n=0}^{N} f_n$ be the partial sums of the series, and let $s$ be its sum.
By linearity of the integral and induction on $N$ , each $s_N$ is Riemann integrable, and $\int_a^b s_N = \sum_{n=0}^{N} \int_a^b f_n$ .
By hypothesis, the series converges uniformly on $[a, b]$ . That is, $s_N \to s$ uniformly on $[a, b]$ .
By steps 2 and 3, the theorem that uniform limits commute with integration applies to $(s_N)$ .
It gives that $s$ is Riemann integrable, and $\int_a^b s_N \to \int_a^b s$ as $N \to \infty$ .
By step 2, $\int_a^b s_N$ is the $N$ -th partial sum of the series $\sum_n \int_a^b f_n$ . By step 5 and the definition of the sum of a series, this series converges, and $\sum_{n=0}^{\infty} \int_a^b f_n(x) \, dx = \lim_{N \to \infty} \int_a^b s_N(x) \, dx = \int_a^b \sum_{n=0}^{\infty} f_n(x) \, dx.$

Idea

Finite sums can be integrated term by term, by linearity. Uniform convergence lets the limit of the partial sums pass through the integral. So the integral of the sum is the sum of the integrals.

Reading path

To understand this page from the ground up, read these 183 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof of the integral of a power, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the reverse triangle inequality, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof that uniform limits commute with integration, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the linearity of differentiation, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the second fundamental theorem of calculus, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the triangle inequality for integrals, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the mean value theorem from Rolle's theorem, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Natural numbers object
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Set definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Ordered pair definition
Continuity at a point definition
Derivative definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
Riemann integrable function definition
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Cartesian product definition
Continuous function definition
Differentiable function definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Riemann integral definition
Subset definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Upper and lower Darboux sums definition
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
Partial sum definition
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Definite integral definition
Tautology definition
Upper and lower Darboux integrals definition
Uniform convergence definition
Convergent series definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Interval definition
Strong induction statement
Logical equivalence definition
Sum of a series definition
Absolute convergence definition
Series of functions definition
Preorder definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Well-ordering principle statement
Double negation law statement
Total order definition
Uniform continuity definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the reverse triangle inequality proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that differentiable functions are continuous proof
Proof that limits preserve non-strict inequalities proof
Proof of the linearity of the integral proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Reverse triangle inequality statement
Differentiable functions are continuous statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Limit of a sum of sequences statement
Linearity of the integral statement
Uniqueness of limits statement
Proof of the monotonicity of the integral proof
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that Darboux and Riemann integrals agree proof
Monotone convergence theorem for sequences statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Proof that a decreasing sequence bounded below converges proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof of the triangle inequality for integrals proof
Bolzano–Weierstrass theorem statement
Product rule statement
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Triangle inequality for integrals statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the power rule by induction proof
Boundedness theorem statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Proof of Fermat's theorem on stationary points proof
Fermat's theorem on stationary points statement
Proof that continuous functions are Riemann integrable proof
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
Proof of the second fundamental theorem of calculus proof
Second fundamental theorem of calculus statement
Proof of the integral of a power proof
Integral of a power statement
Proof that uniform limits commute with integration proof
Uniform limits commute with integration statement
Proof of term-by-term integration proof

Where this leads

1 page builds on this one.

Term-by-term integration

Tag	`PV073ATZBQ`
URI	`m2://PV073ATZBQ`
Kind	proof
Writing	W0 stub
MSC	40A30, 26A42
Proves	Term-by-term integration
Uses	Series of functions, Linearity of the integral, Uniform limits commute with integration, Sum of a series
Changed	4 November 2026

Proof of term-by-term integration

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