A Proof Is an Argument You Can Inspect
In How to Read a Theorem, we identified the domain, hypotheses, and conclusion of a theorem. Reading its proof is the next task: follow the chain of reasoning that shows the conclusion really follows from the hypotheses. A proof is not a collection of calculations that happens to end with the desired claim. Each important step must be supported by a definition, an assumption, a valid inference, or a result established earlier in the course.
When reading, keep two things visible at the same time. The first is the goal: the precise statement that the proof must establish. The second is the available information: the hypotheses and facts already justified. At every step, ask how the new statement follows from that information. If you lose track of the goal, a correct calculation may feel disconnected. If you lose track of the available information, an unjustified leap may seem plausible.
This way of reading also clarifies the difference between understanding a proof and merely checking its final answer. A conclusion can be true while a proposed argument for it is incomplete. Conversely, a line may look unfamiliar but be valid once you identify the definition or theorem that supports it. The aim is to make the dependency between statements explicit.
Read the Opening as a Record of the Proof's Starting Point
The first sentence of a proof often says, “Let \(x\) be an arbitrary real number,” or “Suppose \(n\) is even.” These phrases are not filler. They establish which objects are being considered and which hypotheses may be used. In a proof of a universal conditional statement, the proof typically takes an arbitrary object in the stated domain, assumes the hypothesis, and derives the conclusion. That method was introduced in Introduction to Mathematical Proof.
Once an assumption has been introduced, it may be used throughout the argument. A proof may also introduce an object whose existence is guaranteed by a definition. For example, if \(n\) is even, the definition of evenness gives an integer \(k\) such that \(n=2k\). The integer \(k\) is not an arbitrary extra assumption; its existence is exactly what the definition supplies.
It helps to distinguish an assumption from a conclusion reached later. If the proof begins by supposing \(n\) is even, then \(n=2k\) for some integer \(k\) follows from the definition. But the proof may not simply assume that \(n\) has this form when the theorem has not given evenness. The source of a fact matters.
Rewrite the theorem’s conclusion in a form you can check. If the goal is that an integer is even, the definition tells you to look for an integer \(r\) that expresses it as \(2r\).
Record the domain and hypotheses. Keep them separate from facts derived later in the proof.
For every new equation or claim, identify the definition, earlier line, algebraic rule, or theorem that supports it.
Confirm that the proof has established the stated conclusion, not a nearby or weaker claim.
Follow the Definition Through the Algebra
Definitions often provide the key move in a proof. The proof may begin with a familiar phrase such as “\(n\) is even,” while the conclusion is also phrased in terms of evenness. To check the argument, translate those phrases into the definition and inspect the integer expressions carefully.
Consider the following theorem. The phrase “divisible by \(4\)” means that the integer in question can be written as \(4r\) for some integer \(r\). Thus, the goal is not just to manipulate \(n^2\); it is to produce the required integer factor.
Worked Example: Reading a Proof About the Square of an Even Integer
Theorem. If \(n\) is an even integer, then \(n^2\) is divisible by \(4\).
Proof. Let \(n\in\mathbb Z\) be even. By the definition of evenness, there is an integer \(k\) such that \(n=2k\). Squaring both sides gives $$ n^2=(2k)^2=4k^2. $$ Since \(k\in\mathbb Z\), its square \(k^2\) is an integer. Therefore \(n^2\) has the form \(4r\) with the integer \(r=k^2\), so \(n^2\) is divisible by \(4\). This proves the theorem.
To read this proof, connect each sentence to its role. The opening chooses an arbitrary integer satisfying the hypothesis. The definition of evenness supplies \(k\). The algebra produces the factor \(4\). The fact that \(k^2\) is an integer verifies the remaining condition in the definition of divisibility. The final sentence does not merely repeat the calculation; it matches the result to the target definition.
A common incomplete version stops at \(n^2=4k^2\). The equation is correct, but the reader still needs to know that \(k^2\) is an integer and that this equation has the required form. In this example those steps are straightforward, yet stating them reveals exactly why the definition applies.
Not every line needs a long explanation. A proof may omit routine algebra once the reader can verify it, especially when the omitted calculation is unambiguous. But a proof cannot omit a necessary logical connection. When a line seems too large a jump, expand it for yourself: write the algebra, name the definition, or check the hypotheses of the result being used.
Check Whether a Theorem Is Being Used Correctly
A proof may rely on a theorem established earlier instead of deriving every fact from definitions. That is legitimate, and it is often the clearest approach. To follow such a step, identify the earlier result and check that its hypotheses apply to the objects at hand. A named theorem is not a substitute for checking conditions.
For example, the Triangle Inequality, a later result in this course, says that \(|u+v|\leq |u|+|v|\) for real \(u,v\). A proof that invokes it should make clear what plays the role of \(u\) and \(v\), and that those values are real. The inequality may then be used as a justified step. If the proof instead uses a result about positive numbers, it must also establish that the relevant numbers are positive.
Worked Example: Checking the Hypotheses Behind an Inequality Proof
Theorem. If \(x>3\), then \(x^2>9\).
Proof. Let \(x\in\mathbb R\) and suppose \(x>3\). Subtracting \(3\) gives \(x-3>0\). Also \(x+3>6>0\), because adding \(3\) to \(x>3\) gives \(x+3>6\). The product of two positive real numbers is positive, so $$ (x-3)(x+3)>0. $$ Expanding the left-hand side gives \(x^2-9>0\), and therefore \(x^2>9\). This proves the claim for every real \(x\) satisfying the hypothesis.
When checking the proof, notice that the factorization alone would not establish the result. The sign of both factors is essential. The hypothesis \(x>3\) gives \(x-3>0\), and it also gives \(x+3>0\). Only after establishing both signs can we conclude that their product is positive. The final rearrangement, \(x^2-9>0\) and hence \(x^2>9\), uses the order properties of real numbers.
This also shows why checking the hypotheses matters. If \(x=-4\), then \(x^2=16>9\) happens to be true, but the proof above does not apply: \(-4>3\) is false. A proof establishes its conclusion for the stated inputs; it need not use the same argument to explain every other input for which the conclusion might happen to hold.
Distinguish a Gap from a Different Route
A proof can reach its conclusion through different valid routes. One argument may use a definition directly, while another may use an earlier theorem. Different wording or a different sequence of calculations does not by itself indicate a problem. The essential test is whether each step follows from the permitted starting facts and whether the whole argument reaches the stated goal.
A genuine gap occurs when the argument needs a fact that has not been established. Sometimes the missing fact is easy to supply. Sometimes it is the central difficulty of the theorem. Either way, do not silently insert a needed assumption. For instance, knowing that a product is positive does not, without further information, tell us that both factors are positive: two negative factors also have a positive product. A proof that needs each factor positive must show that separately, or use another valid argument.
| What you see in a proof | Question to ask | What makes the step acceptable |
|---|---|---|
| A substitution from a definition | Does the stated hypothesis meet the definition? | The definition supplies the substituted form and its required domain. |
| An algebraic rearrangement | Do both expressions remain equal or preserve the inequality? | The transformation is valid for the quantities involved. |
| A cited theorem | Are all of the theorem’s hypotheses satisfied? | The objects lie in its domain and meet its conditions. |
| A claim of existence | Has an object been produced or its existence otherwise justified? | The argument establishes an object with every required property. |
| The closing sentence | Is the theorem’s exact conclusion now established? | The final claim matches the target and applies to arbitrary allowed inputs. |
Worked Example: Finding the Missing Direction in an Argument
Consider the claim, “If \(x^2>0\), then \(x>0\),” for real \(x\). A proposed argument says: “Since \(x^2>0\), \(x\) is positive. Therefore \(x>0\).” The last sentence is just a restatement of the conclusion, and the first sentence assumes exactly what needs to be proved. No valid reason is given for concluding that \(x\) is positive from \(x^2>0\).
A direct check confirms that the claim is false. Choose \(x=-2\). Then \(x^2=4>0\), but \(-2\) is not greater than \(0\). This is not merely a gap in this proposed proof; it is a counterexample to the theorem itself. The example teaches two separate reading habits: identify whether each inference is justified, and when a universal claim seems doubtful, test whether a permitted input contradicts it.
Sometimes an apparent gap can be repaired by stating a fact that was left implicit. In the proof about \(x>3\), a reader might ask why \(x+3>0\). The proof supplies the needed comparison: \(x+3>6>0\). That is not an assumption added to the theorem; it is derived from the original hypothesis. By contrast, in the claim about \(x^2>0\), no valid derivation can repair the implication as stated, because the counterexample shows its conclusion need not hold.
Read for Dependencies, Not Just the Final Line
A useful way to keep a proof organized is to make a short dependency record. For each important statement, note whether it comes from the theorem’s hypothesis, a definition, an earlier line, algebra, or a named result. This records the proof’s logical structure without requiring every routine calculation to be rewritten. It also makes it easier to find a particular issue: if a line depends on a fact that appears nowhere in the record, ask whether that fact was established.
The proof of the even-square result has a short dependency chain: evenness gives \(n=2k\); substitution and algebra give \(n^2=4k^2\); the integer property of \(k\) gives \(k^2\in\mathbb Z\); the definition of divisibility by \(4\) gives the conclusion. The proof about \(x>3\) has a different chain: the hypothesis gives two positive factors; positivity gives a positive product; factorization and rearrangement give \(x^2>9\). Seeing these connections is more useful than memorizing either proof word for word.
When a proof is difficult, read it more than once with different questions in mind. First identify the overall strategy and the target. Then check the local steps. Finally, explain the argument in your own words, preserving the assumptions and the reasons for each transition. This is not a demand to invent a new proof; it is a disciplined way to confirm that the written proof actually works.
Check Your Understanding
Use the proof-reading questions from this tutorial to respond to each item.
- In a proof beginning “Let \(n\) be an even integer,” what does the definition of evenness allow the proof to write, and what kind of object must the accompanying \(k\) be?
- In the proof that an even integer’s square is divisible by \(4\), why is it not enough to stop at \(n^2=4k^2\)?
- For the proof that \(x>3\) implies \(x^2>9\), identify why both \(x-3\) and \(x+3\) are positive before their product is used.
- A proof cites a theorem about positive real numbers. What should you check before accepting the application?
- The claim “if \(x^2>0\), then \(x>0\)” is presented without a valid inference. Give a real counterexample and explain what it establishes.
- What four features should you be able to identify when deciding whether a proof is complete?