Writing a Proof Someone Else Can Check
A written proof is a claim about what a reader can verify, not a record of how it was found. Announced structure, named hypotheses and explicit quantifiers are what make it checkable.
What a learner can do afterwards
- Write a proof whose structure is announced in the first line
- Name the hypothesis at the step where it is used, not in a preamble
- Rewrite an argument that hides a quantifier so the order is explicit
1 · Read
A written proof is a claim about what a reader can verify, not a record of how it was found. Announce the structure in the first line: name the method, such as direct, contradiction, or induction, and state the claim. Theo opens well: we prove by contradiction that root 2 is irrational, and suppose root 2 is rational. The reader knows the plan before any deduction starts.
Name each hypothesis at the step where it is used, not only in a preamble. To conclude n is odd from n being prime above 2, write: since n is prime larger than 2, n is odd. A preamble list alone lets uses hide. Likewise write: taking limits, by continuity of f at c, f(xn) converges to f(c). The reader can then check that exact step on the spot.
Make every quantifier explicit and keep its order. Over the integers, for every x there is a y with y above x is true, with y equals x plus 1 as witness. Reversing the order claims one single y above every integer, which is false. If X then Y says X suffices for Y and Y is necessary for X: mixing the directions breaks proofs, especially in if and only if arguments.
State which direction each step proves, and revise by reading the draft cold. Check that each line uses only named facts: a definition, a hypothesis, or an earlier line. When a step feels obvious, spell it out, since obvious to the writer is where readers get lost. A proof that hides a quantifier should be rewritten so the order is explicit.
Announce the plan up front, cite each hypothesis where used, and write quantifiers in order.
2 · Watch
Take it off screen
Where it sits
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.