Proofs in Mathematics: Methods, Examples, and How to Write Them

A mathematical proof is a logical argument, built only from definitions, axioms, and previously established results, that shows a statement is true in every case with no exceptions. Checking examples, no matter how many, is never enough. Proofs use deductive reasoning, and the main techniques are direct proof, proof by contradiction, and proof by induction.

Key Takeaways

• A proof must hold for every possible case; testing examples such as n = 1 through n = 10 can never establish a general claim.
• The polynomial n² − n + 41 produces prime numbers for every integer from n = 1 to n = 40, then fails at n = 41, where the result equals 41².
• Direct proof, proof by contradiction, and proof by induction are the three techniques covered below, each illustrated with a worked example.
• Induction requires two components: a base case and a step showing the statement holds for n+1 whenever it holds for n.
• Formal proofs use rigid symbolic structure, while informal proofs use natural language and are what mathematicians write in practice.

What Is a Mathematical Proof?

A proof is a convincing explanation of why a mathematical statement is true, built only from definitions, axioms, and previously proven theorems or lemmas. A proof relies on deductive reasoning rather than empirical observation, meaning it must account for every case, not just the ones that have been checked.

For example, no amount of checking small values of n can prove that n² − n + 41 is prime for every natural number n. A valid proof must explain why the claim holds universally, with no exception, regardless of how large n becomes.

In mathematical logic, a “formal proof” is a proof written with rigid symbolic structure and little or no natural language. Formal proofs are precise but difficult to read, so in practice mathematicians write informal proofs: arguments expressed in readable prose that remain rigorous enough to be convincing. Any valid informal proof can, in principle, be converted into a formal one, though this conversion is rarely done in practice because of how tedious it is. A large collection of formal proofs is available at Metamath.

Why Are Proofs Necessary Instead of Checking Examples?

Mathematics requires statements that are true in every case, or false, rather than statements that are “usually true.” A pattern that holds for most examined cases is generally uninteresting unless the word “most” is defined with mathematical precision, because checking finitely many cases can never rule out a counterexample among infinitely many possibilities.

The polynomial n² − n + 41 illustrates this danger clearly. It produces a prime number for every integer from n = 1 through n = 40:

Values of n² − n + 41 for n = 1 through n = 10, each a prime number

n	n² − n + 41
1	41
2	43
3	47
4	53
5	61
6	71
7	83
8	97
9	113
10	131

Despite this long run of prime results, the pattern breaks at n = 41, where the polynomial equals 41² rather than a prime number:

$$41^2 – 41 + 41 = 41^2.$$

This single counterexample demonstrates why experimental evidence, however extensive, can never substitute for a proof. Mathematical certainty requires an argument that covers every case, including the ones nobody has checked.

What Are the Main Methods of Proof?

Several proof techniques recur across mathematics, and experience helps in choosing which method fits a given problem. The three most common are direct proof, proof by contradiction, and proof by induction, each described below with a worked example.

How Does a Direct Proof Work?

A direct proof starts from the stated assumptions and proceeds logically, step by step, to the conclusion, without assuming the opposite of what is being proven.

As an example, consider proving that the square of an odd integer is always odd. An odd integer can be written in the form 2n+1. Squaring this expression gives:

$(2n+1)^2 = 4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1.$

This result equals 2m+1 with m = 2n² + 2n, which is the general form of an odd number, so the square of an odd integer is odd.

A second direct proof example: for the linear polynomial aX + b where a ≠ 0, the value −b/a is a root, because substituting it back in gives zero:

$$a\left(\frac{-b}{a}\right) + b = \frac{-ba}{a} + b = -b + b = 0.$$

How Does Proof by Contradiction Work?

Proof by contradiction assumes the negation of the statement to be proven and then derives a logical contradiction from that assumption, which shows the original statement must be true.

Consider the converse of the earlier result: if n² is odd, then n must be odd. To prove this by contradiction, assume instead that n is even, meaning n = 2k for some integer k. Then:

$n^2 = (2k)^2 = 4k^2 = 2(2k^2),$

which means n² is even. This contradicts the starting assumption that n² is odd, so the assumption that n is even must be false. Therefore n must be odd.

Proof by contradiction also appears in everyday reasoning, such as solving a 2×2 mini sudoku where the numbers 1 and 2 must each appear once in every row and column. If the first row already contains a 1, placing a second 1 in that row would violate the rule, so assuming it were possible leads to a contradiction, meaning the remaining cell must contain 2.

How Does Proof by Induction Work?

Mathematical induction proves a statement for all natural numbers, or for all integers greater than or equal to some base value, using a two-part argument similar to toppling dominoes: a base case, and a step showing the statement passes from n to n+1.

Formally, induction requires proving two things: first, that the statement holds for a specific base value a, and second, that for every n ≥ a, if the statement holds for n, it also holds for n+1. Once both parts are established, the statement holds for every integer greater than or equal to a.

As an example, consider proving that 1 + 2 + 3 + … + n equals n(n+1)/2 for every positive integer n. The base case, n = 1, gives 1 = 1(1+1)/2, which checks out. For the induction step, assume the formula holds for n, then:

$$1+2+\dots + n + (n+1) = \frac{n(n+1)}{2} + (n+1) = \frac{n^2+n + 2n + 2}{2} = \frac{(n+1)(n+2)}{2},$$

which shows the formula also holds for n+1, completing the induction.

A second example is Bernoulli’s inequality, which states that (1+x)^n ≥ 1 + nx for every integer n ≥ 0 and every real number x > −1. The base case, n = 0, gives (1+x)⁰ = 1 = 1 + 0·x. For the induction step, assume (1+x)^n ≥ 1 + nx, then multiply both sides by (1+x) ≥ 0:

$$(1+x)^{n+1} \ge (1+nx)(1+x) = 1 + (n+1)x + x^2 \ge 1 + (n+1)x,$$

since x² ≥ 0, which completes the induction step.

How Should a Proof Be Written and Communicated?

Writing a proof clearly is a separate, learnable skill from finding the proof in the first place. The right level of detail depends on the audience: a proof written for high-school students needs more explanation than one written for research mathematicians.

A well-written proof should not consist of symbols alone. Full sentences should guide the reader through each logical step, and calculations should be accompanied by an explanation of what each step accomplishes. Heavy use of formal logic symbols such as ⇒ or ∃ without any surrounding text makes a proof harder to follow. For instance, the following terse proof is technically correct but unfriendly to a reader:

$$n=2k+1 \Rightarrow n^2=(2k+1)^2=4k^2+4k+1 \Rightarrow \exists m: n^2=2m+1$$

A clearer version of the same argument reads: “Let n = 2k + 1. Then n² = (2k+1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, so n² has the form 2m+1 with m = 2k² + 2k, hence it is odd.”

Frequently Asked Questions

What is the difference between a formal proof and an informal proof?

A formal proof has a rigid symbolic structure with little or no natural language, making it precise but hard to read. An informal proof uses natural-language sentences alongside calculations, remaining rigorous while staying readable. Mathematicians write informal proofs in practice, even though any valid informal proof could, in principle, be converted into a formal one.

Why can’t checking many examples prove a mathematical statement?

Checking examples can never rule out a counterexample among the infinitely many cases left unchecked. The polynomial n² − n + 41 produces a prime number for every n from 1 to 40, then fails at n = 41, where it equals 41² instead of a prime, showing why a pattern observed in examples is not a proof.

What are the two required parts of a proof by induction?

A proof by induction requires a base case, showing the statement holds for a specific starting value, and an induction step, showing that if the statement holds for n, it also holds for n+1. Together these two parts establish the statement for every integer at or above the base value.

How does proof by contradiction work?

Proof by contradiction assumes the opposite of what needs to be proven, then shows that this assumption leads to a logical impossibility. Since the assumption cannot hold, the original statement must be true. An example is proving that n must be odd if n² is odd, by assuming n is even and showing that this forces n² to be even, contradicting the premise.

How much symbolic notation should a written proof contain?

A proof should use full sentences to guide the reader through the logical steps rather than relying on symbols such as ⇒ or ∃ alone. Calculations should appear alongside an explanation of what each step accomplishes, since a symbol-only proof, while technically correct, is harder for a reader to follow than one written in clear prose.

Further Reading

The Wikipedia entry on mathematical proof provides additional background and examples of proof techniques.

For guidance on writing mathematical texts clearly, see this guide to mathematical writing.