Basic Analysis I — Real Analysis · Problem 5.1.3

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-integrability

Problem

Let f:[a,b]→Rf : [a,b] \to \mathbb{R} be a bounded function. Suppose there exists a sequence of partitions {Pk}k=1∞\{P_k\}_{k=1}^\infty of [a,b][a,b] such that lim⁡k→∞(U(Pk,f)−L(Pk,f))=0.\lim_{k\to\infty}\bigl(U(P_k,f) - L(P_k,f)\bigr) = 0. Show that ff is Riemann integrable and that ∫abf=lim⁡k→∞U(Pk,f)=lim⁡k→∞L(Pk,f).\int_a^b f = \lim_{k\to\infty} U(P_k,f) = \lim_{k\to\infty} L(P_k,f).

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