Basic Analysis I — Real Analysis · Problem 5.1.7

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-sums

Problem

Suppose

f : [a,b] \to \mathbb{R}

is Riemann integrable. Let

\epsilon > 0

be given. Then show that there exists a partition

P = \{x_0, x_1, \ldots, x_n\}

such that for every set of numbers

\{c_1, c_2, \ldots, c_n\}

with

c_k \in [x_{k-1}, x_k]

for all

k

, we have

\left|\int_a^b f - \sum_{k=1}^n f(c_k)\Delta x_k\right| < \epsilon.