Problem 1
Use the precise definitions, state all hypotheses, apply continuous image of compact/connected set, and verify the conclusion against a boundary case or independent theorem.
Faculty solution guide
Connectedness, Heine–Borel and Continuous Images
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 Connectedness, Heine–Borel and Continuous Images.
Problem 4
The argument must address proving interval and compact-image results with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in functional analysis, data spaces, differential equations, geometry, and optimization, 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.