How to Write Proofs
1 A Proof Is An Organized Explanation
A proof is not a ritual. It is an explanation that starts from assumptions, uses valid steps, and arrives at a clear conclusion.
2 A Reliable Structure
Use this template:
- Restate the claim in your own words.
- Name the assumptions.
- State the proof method.
- Move in small justified steps.
- Mark the exact point where the claim is proved.
3 The Most Common Early Proof Methods
- direct proof
- proof by contrapositive
- proof by contradiction
- induction
- counterexample construction
4 A Small Example
Proposition 1 (Proposition) If \(a\) and \(b\) are even integers, then \(a + b\) is even.
Proof. If \(a\) and \(b\) are even, then there exist integers \(m\) and \(n\) such that \(a = 2m\) and \(b = 2n\). Therefore
\[ a + b = 2m + 2n = 2(m+n). \]
Since \(m+n\) is an integer, \(a+b\) is even.
Notice the structure: assumptions, representation, algebra, conclusion.