Lebl-Basic-Analysis-I · Problem 5.1.7

The Riemann Integral

Mathematics · riemann-sums

Problem

Suppose f:[a,b]→Rf : [a,b] \to \mathbb{R} is Riemann integrable. Let ϵ>0\epsilon > 0 be given. Then show that there exists a partition P={x0,x1,…,xn}P = \{x_0, x_1, \ldots, x_n\} such that for every set of numbers {c1,c2,…,cn}\{c_1, c_2, \ldots, c_n\} with ck∈[xk−1,xk]c_k \in [x_{k-1}, x_k] for all kk, we have ∣∫abf−∑k=1nf(ck)Δxk∣<ϵ.\left| \int_a^b f - \sum_{k=1}^n f(c_k) \Delta x_k \right| < \epsilon.

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