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MathidiaMathematics Academy
Year 1 mathematical synthesis
Lesson 71 of 75
Introduction

Proof synthesis

Objective. Combine definitions and lemmas into a sustained proof.

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

Learn: the key idea

Proof synthesis combines definitions, earlier lemmas and logical connectives into one sustained argument proving a non-trivial statement, rather than applying a single technique in isolation. A typical synthesis might prove a function is continuous (definition), then differentiable (using a limit lemma), then apply the mean value theorem to deduce a global inequality, chaining results that were each proved separately. Good proof synthesis states clearly which earlier result is being invoked at each step and checks its hypotheses are satisfied before use. Skipping hypothesis checks, such as requiring continuity on a closed interval, invalidates the whole chain even if later algebra is correct.

Worked example

Using proof synthesis, combine the facts that f is continuous on [0,2] with f(0)=−1 and f(2)=3 to justify a root of f exists in (0,2).

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Illustrated card showing 0 … 2
  1. 1State the relevant theorem: Intermediate Value Theorem
  2. 2Check hypothesis: f continuous on closed interval [0,2]
  3. 3Note f(0) and f(2) have opposite signs
  4. 4Conclude: a c in (0,2) exists with f(c)=0

Answer: A root exists in (0,2) by the Intermediate Value Theorem

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Common mistake. Quoting a theorem's conclusion without first confirming all its hypotheses hold, for instance applying the mean value theorem to a function that is not continuous on the closed interval required.

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