Sequential Integrability Criterion

A bounded function $f:[a,b]\to \mathbb{R}$ is Riemann integrable iff $\exists \set{P_n}$ a sequence of partitions, such that:

\lim_{n\to\infty}{ \Big[ U(f;P_n)- L(f;P_n) \Big] } =0

In that case:

\int_a^b f = \lim_{n\to\infty} U(f;P_n) = \lim_{n\to\infty} L(f;P_n)

Theorem

Suppose $f$ is Riemann integrable on $[a,b]$ .

\forall \epsilon \gt 0\; \exists \delta \gt 0\; \forall P\; \bigg( \lvert P \rvert \lt \delta \implies \Bigg\lvert \int_a^b f - \sum_{j=1}^n f(\zeta_j)I_j \Bigg\rvert \lt \epsilon \bigg)

where $\zeta_j \in [x_{j-1},x_j], j=1,2,\cdots,n$ .

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October 17, 2026 1 min read

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