A Proof Has a Shape as Well as a Correct Argument
In Building a Chain of Lemmas, we considered how intermediate results can carry an argument from its assumptions toward its goal. A proof also needs an overall structure: the reader must be able to see what is being assumed, what each paragraph establishes, and how the final statement answers the theorem. Organization does not replace valid reasoning, but it makes the reasoning checkable.
A proof is not simply a collection of true sentences about the same subject. Each sentence needs a role in the argument. Some statements introduce arbitrary objects, some record the hypotheses, and others establish facts that will support the conclusion. A calculation may be correct but still leave a gap if it does not explain why its result proves the claim. Conversely, a proof can be concise without being abrupt when its key dependencies are clear.
A useful starting point is to separate the theorem into its logical obligations. For a statement of the form “for every \(x\), if \(P(x)\), then \(Q(x)\),” the proof must address an arbitrary allowed \(x\), assume \(P(x)\), and establish \(Q(x)\). The details of the argument depend on the subject, but those obligations come directly from the statement. The proof should not establish a special case when the claim is universal, nor should it use more assumptions than the theorem supplies.
Plan the Proof Before Writing Its Sentences
Before composing a polished proof, make a compact plan. First write the endpoint in the exact form of the theorem. Then list the assumptions that may be used. Next identify what must be shown and what intermediate facts would make that goal follow. This planning step prevents a common error: finding a plausible calculation and only afterward discovering that it proves a nearby claim rather than the stated one.
For example, a target inequality may depend on a sign condition before multiplication. A proof plan should record both obligations: establish the sign condition, then use it in the comparison. Similarly, an equality claim may need two directions, while a uniqueness claim must compare two objects that satisfy the defining condition. The labels “direct proof,” “proof by cases,” “proof by contrapositive,” and “proof by contradiction” describe different ways to organize reasoning; the appropriate structure is the one that meets the statement’s obligations and keeps its dependencies explicit.
Identify the domain, the quantifiers, the hypotheses, and the precise conclusion.
Choose arbitrary objects where required, and state which hypotheses are now in force.
Decide what intermediate facts or cases would make the endpoint follow.
Establish each fact before using it, and state how it advances the argument.
Make explicit that the final established statement is the theorem’s requested conclusion.
This plan is a working aid, not a mandatory outline to reproduce in every proof. In a short argument, the steps may fit naturally into a few sentences. In a longer proof, sectioning or intermediate claims can make the dependencies easier to follow. The finished proof should reveal enough structure for the reader to verify the argument without having to reconstruct its plan.
Keep Quantifiers and Assumptions in View
The opening of a proof signals what remains in scope. If a theorem says “for every real number \(x\),” begin with an arbitrary real \(x\), not with a specially chosen value. If the statement is conditional, suppose its hypothesis holds. These choices matter: the proof must establish the conclusion for every object covered by the theorem, under exactly the stated conditions.
When a proof introduces a temporary assumption for a case or a contradiction argument, that assumption has a limited scope. A conclusion derived under that temporary assumption cannot be used as though it held unconditionally. The proof must complete the relevant reasoning and then return to the original claim. Likewise, an auxiliary quantity should be defined before it is used, and the proof should state why it belongs to the required domain.
The proof’s final sentence should match the original quantifiers and conditions. If the argument began with arbitrary \(x\) satisfying a hypothesis and derived the required conclusion for that \(x\), state that this proves the assertion for every such \(x\). This is not a ceremonial phrase: it closes the scope opened at the start.
Worked Example: Organizing a Reciprocal Comparison
Theorem. Let \(a,b\in\mathbb R\). If \(0<a<b\), then $$ \frac{1}{b}<\frac{1}{a}. $$
A proof plan should note two facts. The input inequality is \(a<b\), and the intended multiplication must use a positive number so that the order is preserved. The hypotheses give \(a>0\) and \(b>0\), so \(ab>0\) and therefore \(1/(ab)>0\).
Proof. Let \(a,b\in\mathbb R\), and suppose \(0<a<b\). Since \(a>0\) and \(b>0\), their product satisfies \(ab>0\), so \(1/(ab)>0\). Apply the positive-multiplier inequality result from Building a Chain of Lemmas to \(a<b\) with multiplier \(1/(ab)\). It gives $$ \frac{a}{ab}<\frac{b}{ab}. $$ Because \(a\ne0\) and \(b\ne0\), simplifying the two fractions gives \(1/b<1/a\), as required. This proves the theorem.
Notice how the order of the proof makes the sign condition available before multiplication. Starting with the line “multiply by \(1/(ab)\)” without first establishing \(ab>0\) would omit a necessary justification. The proof’s structure exposes that dependency instead of leaving the reader to infer it.
Make Intermediate Claims Earn Their Place
An intermediate claim is useful when it supplies a missing fact, isolates a delicate step, or gives a clear landmark in a longer argument. It is less useful when it merely repeats the previous sentence in different notation. A proof should be organized around meaningful checkpoints rather than a separate heading for every small algebraic move.
When two claims are used together, say how they combine. If an inequality follows by adding two inequalities, identify the pair and use the relevant order property. If a result follows from an earlier lemma, name that result and verify its hypotheses. If a substitution is made, state the values being substituted. These brief explanations show why a step is legitimate, particularly when the same expression could be manipulated in more than one way.
The following theorem uses a short intermediate comparison. Its proof has a distinct sequence: establish that a denominator is positive, compare the numerator with half of that denominator, and then divide by the positive denominator. Keeping those tasks separate helps prevent an invalid division or an unnoticed reversal of an inequality.
Worked Example: A Fraction Below One Half
Theorem. Let \(x,y\in\mathbb R\). If \(0<x<y\), then $$ \frac{x}{x+y}<\frac12. $$
Proof. Let \(x,y\in\mathbb R\), and suppose \(0<x<y\). Since \(x>0\) and \(y>x\), we have \(y>0\). Therefore \(x+y>0\), so division by \(x+y\) will preserve an inequality.
The hypothesis \(x<y\) gives \(2x<x+y\): adding \(x\) to both sides of \(x<y\) gives \(2x<x+y\). Divide both sides by the positive number \(2(x+y)\), or equivalently divide \(2x<x+y\) by the positive number \(2(x+y)\), to obtain $$ \frac{2x}{2(x+y)}<\frac{x+y}{2(x+y)}. $$ The left side simplifies to \(x/(x+y)\), and the right side simplifies to \(1/2\). Thus \(x/(x+y)<1/2\), proving the theorem.
The structure makes the role of each assumption visible. Positivity of \(x+y\) justifies division, while \(x<y\) produces the comparison \(2x<x+y\). Neither fact alone would complete the proof.
Order Cases and Longer Calculations Carefully
A proof by cases should name the cases and establish that they cover every permitted possibility. Within each case, use the assumption belonging to that case and return to the shared conclusion. The method is especially helpful when a definition or expression behaves differently according to a sign or a comparison. A case split is not complete merely because several plausible possibilities have been discussed; the cases must be exhaustive, and each must reach the target.
Longer algebraic arguments also benefit from careful organization. One reliable practice is to avoid jumping directly from a complicated hypothesis to the final expression. Instead, derive small facts whose roles are clear: a factor is positive, a difference is positive, or two expressions are equal. Then use those facts in an order that the reader can follow. In particular, distinguish an equality used to rewrite an expression from an inequality used to compare quantities.
Worked Example: A Polynomial Inequality with a Clear Route
Theorem. For every real number \(t\), if \(t>3\), then $$ t^2-5t+6>0. $$
Proof. Let \(t\in\mathbb R\) and suppose \(t>3\). Then \(t-2>1>0\), and \(t-3>0\). Factor the expression: $$ t^2-5t+6=(t-2)(t-3). $$ Both factors on the right are positive. The product of two positive real numbers is positive, so \((t-2)(t-3)>0\). By the displayed equality, \(t^2-5t+6>0\), which is the required conclusion. This proves the theorem.
The proof is organized around the factorization rather than a string of unexplained expansions. The factorization identifies the two sign checks needed; the hypothesis supplies each one; and the equality then transfers the positive product to the original expression. Had the proof only stated that the polynomial factors, it would still need to explain why that factorization establishes positivity.
Revise for Validity and Readability
A proof review should check the logical route and the presentation separately. For validity, check that each object is in its stated domain, every hypothesis is available when used, and each inference follows from a definition or an established result. Confirm that any division uses a nonzero quantity and that multiplying an inequality uses a multiplier of the appropriate sign. These are checks of mathematical substance.
For readability, check whether the reader can tell what each paragraph is doing. A sentence such as “this is positive” may be ambiguous if several expressions have just appeared. Name the quantity. A phrase such as “by the lemma” is useful only if the cited result applies to the values at hand; make any substitution or hypothesis check explicit when it is not immediate. Remove exploratory calculations that did not contribute to the finished proof, but retain the short explanations needed to connect the successful steps.
The organization should not disguise a gap. A list of correct equations is not automatically a proof if the transition from one line to the next depends on an unstated condition. Nor does a polished paragraph repair an invalid inference. Clear structure makes it easier to locate such problems: each step has a purpose, and the reader can compare that purpose with the evidence supplied.
Check Your Understanding
For each question, focus on the role and order of the proof’s statements.
- For a theorem stating “for every real \(x\), if \(P(x)\), then \(Q(x)\),” what should the proof establish at its opening?
- In the reciprocal comparison, why is the positivity of \(1/(ab)\) established before multiplying \(a<b\) by that quantity?
- In the fraction example, identify which hypothesis gives \(2x<x+y\) and which fact justifies dividing by \(2(x+y)\).
- In the polynomial example, what two sign facts must be checked after factoring \(t^2-5t+6\)?
- Why should a temporary assumption in a case or contradiction argument not be treated as an unconditional fact?
- Name one check that concerns mathematical validity and one check that concerns readability when revising a proof.