Mathematical Proof

Proof by Deduction

Use logical steps from known facts to reach the conclusion.

Example: Prove $(n+1)^2 - (n-1)^2 = 4n$.

LHS = $n^2 + 2n + 1 - (n^2 - 2n + 1) = 4n$ = RHS ✓

Proof by Exhaustion

Check all possible cases.

Example: Prove $n^2 + n + 1$ is odd for $n = 1, 2, 3$: $3, 7, 13$ — all odd ✓.

Proof by Counter-Example

Disprove a statement by finding one case where it fails.

Example: "$n^2 > n$ for all integers." Counter: $n = 0$, $0^2 = 0 \not> 0$ ✗.

Proof by Contradiction

Assume the opposite, show this leads to a contradiction.

Example: Prove $\sqrt{2}$ is irrational.

Assume $\sqrt{2} = \frac{p}{q}$ (in lowest terms). Then $2q^2 = p^2$, so $p$ is even. Let $p = 2k$: $2q^2 = 4k^2$ → $q^2 = 2k^2$, so $q$ is even. Contradiction: both $p$ and $q$ are even, so not in lowest terms.