Essential Proof Techniques in Higher Mathematics

Mathematical proofs are the backbone of rigorous mathematics. Unlike computation, which finds answers, proof establishes truth. This guide covers the most important proof techniques you will encounter in advanced mathematics courses.

Direct proof is the most straightforward approach. To prove P implies Q, assume P is true and logically derive Q. For example, to prove that the sum of two even numbers is even: if a equals 2m and b equals 2n, then a plus b equals 2(m plus n), which is even.

Proof by contradiction assumes the negation of what you want to prove and derives a contradiction. To prove there are infinitely many primes, assume finitely many: p_1, p_2, ..., p_n. Then N equals p_1 times p_2 times ... times p_n plus 1 is either prime itself or divisible by a prime not in the list, contradicting the assumption that the list was complete.

Mathematical induction proves statements for all natural numbers. The base case verifies the statement for n equals 1. The inductive step assumes the statement holds for n equals k (the inductive hypothesis) and proves it for n equals k plus 1. For example, the sum of the first n positive integers equals n(n plus 1) over 2. The base case: 1 equals 1(2)/2 equals 1, verified. The inductive step: assume the sum to k equals k(k plus 1)/2. The sum to k plus 1 equals k(k plus 1)/2 plus (k plus 1) equals (k plus 1)(k plus 2)/2, completing the induction.

Proof by contrapositive proves P implies Q by proving the equivalent statement not-Q implies not-P. This is often easier when the conclusion contains negations. For example, to prove that if n squared is even then n is even, prove the contrapositive: if n is odd then n squared is odd. If n equals 2k plus 1, then n squared equals 4k squared plus 4k plus 1 equals 2(2k squared plus 2k) plus 1, which is odd.

Existence proofs can be constructive (exhibiting an example) or non-constructive (showing an example must exist without finding it). The intermediate value theorem is a non-constructive existence proof: it guarantees a root exists without telling you where it is.

Uniqueness proofs establish that at most one object satisfies given conditions. The standard approach is to assume two objects satisfy the conditions and prove they must be equal.

Counterexamples disprove universal claims. A single counterexample suffices to show a statement is false. For instance, the claim that all continuous functions are differentiable is disproved by the absolute value function, which is continuous everywhere but not differentiable at zero.