Basic Analysis I — Real Analysis · Problem 5.1.10

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · darboux-sums

Problem

Let

f : [0,1] \to \mathbb{R}

be a bounded function. Let

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

be a uniform partition of

[0,1]

, that is,

x_i = i/n

. Is

\bigl\{L(P_n, f)\bigr\}_{n=1}^\infty

always monotone? Yes/No: Prove or find a counterexample.