Problem 1
Show s is an upper bound and every smaller number fails to be an upper bound.
Faculty solution guide
Bounds, Suprema and Infima
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 Bounds, Suprema and Infima.
Problem 4
The argument must address proving upper-bound properties and least-upper-bound conclusions with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in calculus, numerical analysis, optimization, approximation, and advanced analysis, 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.