Open a maths textbook and flip to any theorem. You will typically find two things underneath it: a proof, and a handful of worked examples. Both have neat algebra. Both finish with a boxed result. Both feel persuasive. And both were probably the first pages you skipped when cramming for a test. The question is fair: "If I have read the worked example and it checks out, do I still need the proof? Aren't they the same thing?"
They are not. They live in different epistemic categories and they do different jobs. Mixing them up is one of the oldest student confusions in mathematics.
What each one is
Worked example
A worked example is the demonstration of a technique or claim applied to a particular set of values. It shows how the procedure plays out on specific numbers, so the reader can imitate it.
Proof
A proof is a chain of logical steps that establishes a statement is true for every object in its scope, using only definitions, axioms, and previously proved results.
A worked example is a demonstration. A proof is an argument.
The verb test
One fast way to tell them apart: look at the verbs.
A worked example runs on verbs like compute, substitute, plug in, evaluate, simplify, check. The activity is procedural. Something happens to specific numbers, and you watch it happen.
A proof runs on verbs like assume, by definition, it follows, therefore, hence, by the previous theorem. The activity is deductive. Statements are joined by logical connectives, and each step is justified.
Why this verb pattern differs: a demonstration needs to show you how the procedure looks when executed; an argument needs to show you why the conclusion has to be true. Computation and deduction are different linguistic modes, and the verbs track the difference.
Same algebra, different roles
Consider "the sum of two even integers is even."
Worked example. Take m = 4 and n = 6. Then m + n = 10, which is even. So the claim holds for this pair.
Proof. Let m and n be arbitrary even integers. Then m = 2a and n = 2b for some integers a, b. Compute:
Since a + b is an integer, m + n is of the form 2 \cdot (\text{integer}), so m + n is even. \blacksquare
The worked example took two specific numbers (4 and 6) and watched what happened. The proof took two arbitrary even integers and derived the conclusion from the definition. The worked example covers one case. The proof covers all cases — including m = 4, n = 6, which is why the worked example agreed with it.
A picture of the coverage gap
Why worked examples do not replace proofs
Imagine someone claims "n^2 + n + 41 is prime for every integer n \geq 0." You try n = 0: you get 41, prime. Try n = 1: 43, prime. Try n = 2: 47, prime. Try n = 5: 71, prime. After ten successful worked examples you might believe the claim — and you would be wrong. At n = 40, the expression equals 40^2 + 40 + 41 = 41 \cdot 41, which is not prime. (This is a genuine trap known as Euler's prime-producing polynomial.)
Ten worked examples that support a claim do not prove the claim. An infinite number of worked examples that support it do not prove it either — because you can never list infinitely many. Only an argument that covers every case simultaneously proves the claim. See Why 20 Examples Aren't Enough for a longer discussion.
Why proofs benefit from worked examples
Despite their different roles, the two are natural partners. A good textbook will present:
- The proof first (or the theorem statement followed by the proof) — establishing that the claim is true.
- One or more worked examples after — showing what the claim looks like on particular inputs.
The proof does the heavy lifting of certification. The worked example builds your intuition about how to use the theorem in practice. You read the proof to become convinced; you read the worked examples to become fluent. Both are necessary, and neither substitutes for the other.
A useful analogy: in a physics class, the derivation of the projectile equation is the proof; the numerical problem "a cricket ball is hit at 30°..." is the worked example. The derivation tells you the equation works for every angle; the worked example shows you how to plug in numbers. You need both to do physics.
A side-by-side comparison table
| Feature | Worked example | Proof |
|---|---|---|
| Scope of conclusion | One specific case | All cases in the statement |
| Use of variables | Replaced by numbers | Kept abstract, arbitrary |
| Justification style | "compute," "check" | "by definition," "therefore" |
| Failure mode | Does not cover other cases | A single wrong step invalidates |
| Produces | Intuition, fluency | Certainty, universal claim |
| Ends with | Numerical answer | \blacksquare or QED |
If the piece of writing in front of you uses specific numbers and ends with a numerical answer, it is a worked example. If it uses variables kept abstract and ends with a claim of general truth, it is a proof.
The trap: presenting a worked example as a proof
A very common student error: writing "Let n = 2. Then n^2 = 4, which is even. QED." This is a worked example dressed in the costume of a proof. The \blacksquare at the end does not make it a proof; the coverage does. A proof must cover every even n, not just one.
The correction is to keep the variable abstract. Write "let n be an arbitrary even integer, so n = 2k for some integer k," and carry the k through the algebra. The abstract variable is what gives the proof its universal scope. See What Does Arbitrary Add?.
When a worked example is a proof
One special case deserves a footnote. For existential claims — statements of the form "there exists an x such that..." — a single worked example is a proof. If the claim is *"there exists an integer n such that n^2 - 5n + 6 = 0,"* then exhibitingn = 2and computing4 - 10 + 6 = 0$ is a complete proof. You needed one example; you produced one example; the claim is established.
But for universal claims — "for every n..." — no finite collection of examples suffices. Most of the theorems you meet in school are universal claims, which is why worked examples almost always support proofs rather than replace them. The existential/universal distinction is the one you want to keep in mind.
The short summary
- Worked example = demonstration on a specific case. Builds fluency; does not certify universal claims.
- Proof = argument covering every case. Certifies the claim is true always.
- Same algebra can appear in both; the scope of what is established is what differs.
- Ten worked examples do not make a proof of a universal claim.
- Existential claims ("there exists") are the exception — one example is a proof.
- Read proofs to become convinced; read worked examples to become fluent. You need both.
The next time you finish reading a theorem, check whether the paragraph that follows keeps the variables abstract (proof) or plugs in numbers (example). If it is only the example, the theorem is believed, not yet established. For certainty, you need the proof.
Related: Mathematical Proof — Direct Proof · Why 20 Examples Aren't Enough · 100 Examples vs One Direct Proof · What Does Arbitrary Add?