Proof Technique and Rigorous Argumentation
This domain tests the standards a written proof must meet: correct quantifier order, justified transitions, valid induction, awareness of converse and contrapositive, and appropriate use of counterexamples. Graders reward explicit hypotheses checks and penalize gaps hidden by notation.
Proof structures
Direct proof assumes the hypotheses and derives the conclusion. Contrapositive proof establishes not Q implies not P, which is equivalent to P implies Q. Contradiction assumes P and not Q and derives an absurdity. Equivalence requires both directions. Set equality requires double inclusion. Uniqueness assumes two objects satisfying the property and proves they coincide. Induction requires a base case, an inductive hypothesis, and a step that uses it; strong induction allows all cases up to n.
ε-arguments
Order of choices is part of the logic. To prove 'for every ε > 0 there exists δ > 0 with property P(ε, δ)', fix an arbitrary ε first, then construct δ, typically as a function of ε and of the fixed data. To disprove a universal claim, one counterexample suffices; to disprove an existence claim, every candidate must be shown to fail.
Worked example
Bernoulli's inequality: (1 + x)^n ≥ 1 + nx for x ≥ −1 and integers n ≥ 0. Base case n = 0: both sides equal 1. Inductive step: assume (1 + x)^n ≥ 1 + nx. Because x ≥ −1, we have 1 + x ≥ 0, so multiplying preserves the inequality: (1 + x)^{n+1} ≥ (1 + nx)(1 + x) = 1 + (n + 1)x + nx² ≥ 1 + (n + 1)x. The hypothesis x ≥ −1 is used precisely at the multiplication step; without it the argument collapses.
Traps
Reversing an implication; assuming the conclusion; using 'without loss of generality' without a genuine symmetry; omitting the base case or failing to use the inductive hypothesis; presenting one example as a proof; failing to verify theorem hypotheses such as compactness, continuity, or nonzero determinant; concluding 'for all ε' from 'for some ε'; dividing by a quantity that may vanish.