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MathidiaMathematics Academy
Mathematical research methods
Lesson 68 of 75
Standard practice

Proof design

Objective. Plan lemmas and select proof architecture.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Proof design is the planning stage before writing a proof: choosing an overall architecture (direct argument, contradiction, contrapositive, induction) and identifying the lemmas needed as stepping stones to the main result. For example, to prove √2 is irrational, the architecture of choice is proof by contradiction: assume √2 = p/q in lowest terms, derive that both p and q must be even, contradicting lowest terms. For an induction proof, the design must specify the base case, the inductive hypothesis, and exactly how the inductive step uses that hypothesis, since an incomplete inductive step invalidates the whole argument even if the base case is correct.

Worked example

For a proof design question, which proof architecture is most appropriate for showing that there is no largest prime number?

Options
  • Direct computation of all primes
  • Proof by contradiction assuming a largest prime exists and constructing a new prime
  • Proof by induction on the number of primes
  • Proof by exhaustive case checking of small numbers
  1. 1Consider the structure of the claim: it asserts no finite maximum exists.
  2. 2Direct computation and case checking cannot cover infinitely many numbers.
  3. 3Assuming a largest prime p exists and constructing a number with a prime factor greater than p gives a contradiction.
  4. 4This contradiction architecture is the standard, appropriate proof design for this claim.

Answer: Proof by contradiction assuming a largest prime exists and constructing a new prime

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Common mistake. A common mistake in proof design is choosing induction and writing the inductive step without ever verifying the base case, leaving the argument logically incomplete even though the step itself may be valid.

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