Lebl-Basic-Analysis-I · Problem 5.5.8

The Riemann Integral

Mathematics · improper-integrals-cauchy-criterion

Problem

Suppose f:[0,∞)→Rf : [0, \infty) \to \mathbb{R} is Riemann integrable on every interval [0,b][0, b]. Show that ∫0∞f\int_0^{\infty} f converges if and only if for every ϵ>0\epsilon > 0 there exists an MM such that if M≤a<bM \leq a < b, then ∣∫abf∣<ϵ\left|\int_a^b f\right| < \epsilon.

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