---
title: "Proof by Deduction and Exhaustion"
description: "Argue from what is given to what is claimed with every step justified, and settle a claim about a small finite set by checking every case in it."
canonical: https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f
source: https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f.md
retrieved: 2026-09-12
---

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# Proof by Deduction and Exhaustion

Argue from what is given to what is claimed with every step justified, and settle a claim about a small finite set by checking every case in it.

Subject: Mathematics · Area: Mathematical Thinking · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f

## Ready when they can

- Prove that the sum of two odd numbers is even by writing each as 2n + 1
- Prove a claim about the integers 1 to 5 by checking each case
- Explain why three worked examples do not prove a general statement

## Lesson: Small certain steps, every case checked

Deduction chains small certainties into big ones, and each link cites a definition, never a hunch. Write an odd number as 2n + 1, add two of them, and factor: (2n + 1) + (2m + 1) = 2(n + m + 1). The sum is twice an integer, which is exactly what even means.

**Example.** Exhaustion works when the claim covers a small finite set you can meet face to face. To prove n squared + n is even for n = 1 to 5, check each door: 2, 6, 12, 20, 30, all even. List the set first so nothing hides, then verify every member.

Three worked examples prove nothing about all n, since infinitely many integers remain unchecked. A proof must handle an arbitrary n, using its definition rather than its value. If any sentence needs trust, split it finer until a sceptic can bite each link.

**Tip.** Splitting {1, 2, 3, 4, 5} into odds and evens can shorten exhaustion, but the blocks must rejoin to the whole. A case split that drops 5 proves nothing about 5. Cover everything, conquer each block, and confirm the cover.

**Recap.** Cite a definition for every deductive link, and check every case an exhaustion claim covers.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Proving Geometry Facts with Vectors](https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf)
- [Algebraic Proof with Odd, Even and Consecutive Numbers](https://lightmysky.com/learn/mathematics/algebraic-proof-with-odd-even-and-consecutive-numbers-mt_p9bajOC17q)
- [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ)

## Opens up

- [Loop Invariants and Proving an Algorithm Correct](https://lightmysky.com/learn/computing/loop-invariants-and-proving-an-algorithm-correct-mt_fWnstju2ns)
- [Proof by Contradiction and Counterexample](https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8)
- [Statements, Quantifiers and Negation](https://lightmysky.com/learn/mathematics/statements-quantifiers-and-negation-mt_S5NvyPxEZ_)
- [Vector Spaces and Subspaces](https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n)