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MathidiaMathematics Academy
Further Mathematics synthesis
Lesson 73 of 75
Standard practice

Advanced proof synthesis

Objective. Build multi-stage exact arguments with clear dependencies.

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

Learn: the key idea

Advanced proof synthesis builds multi-stage arguments that combine several proof techniques, such as induction, direct algebraic proof, and proof by contradiction, into one coherent chain of reasoning with every step's logical dependency made explicit. Such a proof might use induction to establish a general formula, then use that formula within a direct argument to prove an inequality or limiting behaviour. Clarity about what is assumed, what is being proved, and how each stage depends on the previous one is essential to avoid circular or incomplete reasoning.

Worked example

Prove that for all positive integers n, the inequality 2ⁿ > n holds, using proof by induction as part of a synthesis of proof techniques.

2
Illustrated card showing 2
  1. 1Base case n=1: 2¹=2 > 1, true
  2. 2Assume 2^k > k for some positive integer k
  3. 3Then 2^(k+1) = 2×2^k > 2k, since 2^k>k implies doubling preserves the inequality
  4. 4Since k≥1, 2k = k+k ≥ k+1, so 2^(k+1) > k+1, completing the induction

Answer: Proved: 2ⁿ > n for all positive integers n

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Practice

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Common mistake. Students write a long proof but skip stating which earlier result or assumption justifies each new line, making the logical structure impossible to follow or check. Always reference exactly which previous step or known result justifies each new line of a multi-stage proof.

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