Basic Analysis I — Real Analysis · Problem 5.5.9

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · improper-integrals

Problem

Suppose f:[0,∞)→Rf : [0, \infty) \to \mathbb{R} is nonnegative and decreasing. Prove: a) If ∫0∞f<∞\int_0^{\infty} f < \infty, then lim⁡x→∞f(x)=0\lim_{x \to \infty} f(x) = 0. b) The converse does not hold.

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