Problem 1
For every ε>0 there is a partition P with U(f,P)-L(f,P)<ε.
Faculty solution guide
Properties of the Riemann Integral
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Properties of the Riemann Integral.
Problem 4
The argument must address proving linearity, monotonicity, and estimates with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in accumulation, probability, physics, engineering, and numerical quadrature, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.