Problem 1
Use the precise definitions, state all hypotheses, apply goal decomposition, and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Proof Planning, Backward Analysis and Final Presentation
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 Proof Planning, Backward Analysis and Final Presentation.
Problem 4
The argument must address working backward privately while presenting a clean forward argument with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in algebra, number theory, inequalities, algorithm correctness, and structural mathematics, 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.