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A-Level Mathematics • Proof • A3

Proof by Exhaustion

Learn how to prove a statement by checking every possible case in a finite set.

Video Lesson 12 Minutes Pure Mathematics Advanced Level

🎬 Animated Lesson

🎙 Narration

See how proof by exhaustion divides a problem into a finite number of possible cases and verifies each one.

🧠 Key Idea

A proof by exhaustion is valid only when every possible case has been considered.

✅ Learning Outcome

After completing this lesson you should be able to:

  • Recognise when exhaustion is appropriate
  • List all possible cases systematically
  • Verify the statement in each case
  • Write a complete conclusion

Worked Example

Prove that n² + n is even for n = 1, 2, 3 or 4.

Check each possible value:

n = 1: n² + n = 1 + 1 = 2, which is even.

n = 2: n² + n = 4 + 2 = 6, which is even.

n = 3: n² + n = 9 + 3 = 12, which is even.

n = 4: n² + n = 16 + 4 = 20, which is even.

Every possible case is even, so the statement is true.

Quick Quiz

When is proof by exhaustion most suitable?

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