Proof Techniques Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proof Techniques.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Proof techniques are standard strategies for establishing mathematical claims under different structures.

Choose the argument tool that matches the claim type and assumptions.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Different statements are best proved with different methods.

Common stuck point: Students use contradiction when a direct route is simpler, or vice versa.

Sense of Study hint: Classify the claim first (implication, universal, recursive) before selecting a method.

Worked Examples

Example 1

easy
Name four proof techniques, give a one-sentence description of each, and identify which is best suited to prove: 'For all n \ge 1, 3 \mid (n^3 - n).'

Solution

  1. 1
    1. Direct proof: assume the hypothesis and derive the conclusion by logical steps.
  2. 2
    2. Proof by contradiction: assume the negation of the goal and derive a contradiction.
  3. 3
    3. Proof by contrapositive: prove \neg q \Rightarrow \neg p instead of p \Rightarrow q.
  4. 4
    4. Mathematical induction: prove a base case and an inductive step for statements indexed by \mathbb{N}.
  5. 5
    Best technique for 3 \mid (n^3-n): direct proof. Factor: n^3-n = n(n-1)(n+1) — three consecutive integers, so one is divisible by 3. Done.

Answer

n^3-n=n(n-1)(n+1);\text{ direct proof via consecutive integers works best}
Knowing multiple proof techniques and choosing the most efficient one for a given claim is a key mathematical skill. Factoring n^3-n reveals the consecutive-integer structure, making a direct proof immediate.

Example 2

medium
Compare direct proof and proof by contrapositive for: 'If n^2 is even, then n is even.' Which technique is more natural here?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which proof technique is most appropriate for: 'There exists a real number x such that x^2 = 2'? Apply it.

Example 2

medium
Prove using mathematical induction: 3^n > 2n+1 for all n \ge 2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

proof intuitioncontrapositivequantifiers