--- title: "Writing a Proof Someone Else Can Check" description: "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." canonical: https://lightmysky.com/learn/mathematics/writing-a-proof-someone-else-can-check-mt_nZ-zMzj18N source: https://lightmysky.com/learn/mathematics/writing-a-proof-someone-else-can-check-mt_nZ-zMzj18N.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Mathematical Thinking · Ages 22 to 23 Page: https://lightmysky.com/learn/mathematics/writing-a-proof-someone-else-can-check-mt_nZ-zMzj18N ## Ready when they can - Announce a proof's claim and method in its first line, order deductions by their dependencies, and identify circular support - State the hypotheses at the start and cite each one at the step where it is used - Make quantifier order and dependent choices explicit, and state the negation of a quantified claim ## Lesson: Writing proofs a stranger can verify When reading a paper, you identified its target and hypotheses. Give your reader the same starting point. Open with the claim and method, such as direct proof, contradiction or induction. In a proof by contradiction, assume the negation of the target claim. Announcing this plan does not prove the claim; the deductions must still do that. Arrange deductions so their supporting facts come first. Introduce each object and its domain before using it. For example, if m is divisible by 5, write m = 5k for some integer k. Justify each inference with a definition, hypothesis, established theorem or earlier line. Assuming the desired conclusion and repeating it does not derive it from the hypotheses. *(drawing: Each deduction must use facts already available to the reader.)* **Example.** State the hypotheses at the start, then cite each one where it supports a deduction. A prime p > 2 is odd: an even p > 2 would have 2 as a divisor other than 1 or p. Give this reason at the parity step. Division needs its own condition: from ha = hb and h nonzero, divide by h to obtain a = b. The equality alone does not justify division by h. Write x_n for the nth term of a sequence in f's domain. If x_n tends to c and f is continuous at c, then f(x_n) tends to f(c). Cite convergence and continuity at this step. Continuity at c alone does not establish the conclusion for inputs tending elsewhere. Inputs outside the domain do not supply values of f. Name the domains and preserve quantifier order. For every integer x there is an integer y with y > x: after x is given, choose y = x + 1. There is no single integer y greater than every integer x. For any proposed fixed y, taking x = y makes y > x fail. *(drawing: Here the choice of y may depend on the x already given.)* Saying that each student has solved a problem allows the problem to depend on the student. Saying that there is a problem every student solved requires a common choice. Check each student's solved problems, then look for one shared by all. A common upper bound follows this order: choose M once, then check every allowed input against M. Swapping mixed quantifiers can change a claim; two universal quantifiers can exchange order. To negate 'for every a there is a b for which the relation holds', write 'there is an a for which every b makes the relation fail'. Keep that exceptional a fixed while checking all possible b. To refute 'every file has a readable comment', exhibit a file with no readable comment. One unreadable comment is not enough if the file also has a readable one. If X then Y makes X sufficient for Y and Y necessary for X. This implication alone does not establish if Y then X. An if and only if claim needs both implications, each justified. Label the direction being proved. When revising, check the support for each inference: a definition, hypothesis, established theorem or earlier line. Writing 'obvious' beside a gap does not supply support. **Recap.** Announce the plan up front, cite each hypothesis where used, and write quantifiers in order. *This lesson, practice bank and flashcards were written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Reading a Mathematics Paper](https://lightmysky.com/learn/mathematics/reading-a-mathematics-paper-mt_pTF-HzLWE6) ## Optional connections - [Statements, Quantifiers and Negation](https://lightmysky.com/learn/mathematics/statements-quantifiers-and-negation-mt_S5NvyPxEZ_) ## Opens up - [Definitions as Design Decisions](https://lightmysky.com/learn/mathematics/definitions-as-design-decisions-mt_b6ip8eP0-d)