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Completeness and proof language precede sequences, topology, and continuity.

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Expected preparation: Introduction to proofs; Single variable calculus

Prompt 1Proof language, sets, functions, and quantifiers
Negate a nested quantified claim and construct a valid witness. Explain your reasoning and identify any hypothesis that the conclusion needs.
Private scratchpad — not submitted or graded
Prompt 2Ordered fields, bounds, and completeness
Prove the epsilon characterization of a supremum or infimum and apply it without a limit theorem. State every assumption you use and give a complete argument.
Private scratchpad — not submitted or graded
Prompt 3Sequences, Cauchy criteria, and numerical series
Prove a Cauchy sequence with a convergent subsequence has the same limit and locate completeness precisely. State every assumption you use and give a complete argument.
Private scratchpad — not submitted or graded