Here is a proof-looking paragraph:
"Suppose n is even. Then n^2 is even. Since n is even, n = 2k, so n^2 = 4k^2, which is clearly even. Therefore n^2 is even."
At first glance it looks fine. Read it slowly and you realise several lines are doing nothing. "Then n^2 is even" appears before it is established — it is just the conclusion restated. "Which is clearly even" leans on the word clearly instead of a reason. "Therefore n^2 is even" appears twice, front and back, but the justification in the middle is thin.
The discipline that fixes this is blunt: every line in your proof must have a reason, and you must be able to name the reason. If you cannot name the reason — if the line just sounds good or fills space or repeats the previous line with different words — delete the line. A proof made of three justified lines is stronger than a "proof" made of nine unjustified ones.
The two questions every line must pass
Before keeping any line in a proof, ask it two questions:
- Why is this line true? — What rule, definition, algebra step, or prior line justifies it?
- What does this line contribute? — What is the new fact this line gives us that the previous lines did not?
A line that cannot answer both is a line that does not belong.
Why both questions matter: question 1 catches lines that sound true but have no support — they are the "clearly," "obviously," "it is known" lines that hide gaps. Question 2 catches lines that are true but add nothing — they just restate what you already had, burning space without advancing the argument. Good proofs are dense in both senses: every line is justified and every line is new.
The line audit
The cleanest way to practise this is to take a proof you have written, number each line, and next to each line write two things: the reason and the new fact. Watch what happens.
Here is the bad version from the opening, audited:
| # | Line | Reason | New fact |
|---|---|---|---|
| 1 | Suppose n is even. | assumption | hypothesis stated |
| 2 | Then n^2 is even. | — | (restates conclusion without support) ✗ |
| 3 | Since n is even, n = 2k. | definition of even | algebraic form of hypothesis |
| 4 | So n^2 = 4k^2. | substitution + squaring | algebra step |
| 5 | Which is clearly even. | — | (hand-wave, no justification) ✗ |
| 6 | Therefore n^2 is even. | — | (repeats line 2) ✗ |
Lines 2, 5, and 6 have no reason-column. Strike them. Here is the proof after the audit:
"Suppose n is even. Then n = 2k for some integer k (definition of even), so n^2 = (2k)^2 = 4k^2 = 2(2k^2). Since 2k^2 is an integer, n^2 = 2m for m = 2k^2 \in \mathbb{Z}, which by definition means n^2 is even. \blacksquare"
Every line now has a reason. Every line advances the argument. And the proof is shorter, not longer, because the junk lines are gone.
The visualisation: red ink on unjustified lines
Drag the slider to watch three common classes of bad lines appear in red. The green lines are the ones that pass both audits and make it into the final proof.
The three most common bad lines
Three patterns show up again and again in student proofs. Learn to spot them.
1. The hand-wave
"This is clearly …", "obviously …", "it is easy to see that …", "as everyone knows …".
When you write any of these phrases, stop. Ask yourself: if it is clear, can I say why? If the answer is "because of [specific reason]," replace the word clearly with the actual reason. If the answer is "I don't know, it just feels right," you have a gap in the proof — and gaps do not get patched by confidence.
Why clearly is a red flag: a reader cannot check "clearly." They can check "by the definition of even, a number of the form 2k is even." The word clearly is a promise that you won't show the work; a proof is where you do show the work.
2. The premature conclusion
This is when you announce the conclusion near the top of the proof, then prove it, then announce it again at the bottom. The opening announcement is not a proof step — it is a preview — and previews do not belong inside the proof. The preview goes in the statement of the theorem; the proof starts with the hypothesis and ends with the conclusion. No peeks in between.
3. The restatement
"Since n is even, n is even." "So we have n = 2k, which means n = 2k." "n^2 = 4k^2. In other words, n^2 = 4k^2." These lines feel helpful but contribute nothing — they use different words to say the same thing. Every line must contribute a new fact, not paraphrase an old one.
The only legitimate version of restatement is the targeted summary: "We have established n^2 = 2m with m \in \mathbb{Z}; by definition, n^2 is even." The second clause here is not a restatement — it is the application of a definition, which is a genuine new step (you are invoking the definition of even to re-encode the equation into the conclusion's language).
The positive habit
As you write each line of a proof, write a two-word tag for its reason in a margin:
- "by def" (by definition)
- "by algebra" (by algebraic manipulation)
- "by hyp" (by hypothesis)
- "by thm X" (by a theorem you are invoking)
- "by substitution"
- "by closure" (integers are closed under +, ×, etc.)
If you cannot tag a line with one of these (or a similar justification), the line has no reason — cut it. Once the proof is complete, you can erase the margin tags; they are scaffolding, not part of the final artefact.
This habit connects directly to the "why" annotations you see in polished proofs on padho-wiki — those <span class="why">Why: ...</span> boxes are the visible version of the margin tags. Every good proof has this internal commentary, even if it is eventually erased for brevity.
When "I feel it is true" is a warning sign
Sometimes you write a line because it feels right but you cannot immediately name the reason. That is fine — provisionally. Mark the line with a question mark or a note: "[needs justification]." Continue the proof. When you come back to review, you must fill in the justification or remove the line. A line that survives review without a reason is a bug waiting to bite you on the exam.
A particularly dangerous case is when you rely on a previous result you have not clearly stated. "Since the square of an odd number is odd, …" is only a legal line if you have proved this already or it is a known theorem the reader can look up. If it is neither, you cannot use it; either prove it first or use a different approach.
The short summary
- Every line in a proof must answer "why is this true?" If you can name the reason, keep the line. If you cannot, cut it.
- Three most common bad lines: hand-waves ("clearly"), premature conclusions (restating the theorem before you've proved it), and restatements (same fact in different words).
- Mark each line's reason in the margin as you write: by definition, by algebra, by hypothesis, by theorem. This forces the audit to happen live.
- A three-line proof where every line is justified is stronger than a ten-line "proof" with gaps.
- "[needs justification]" is an acceptable temporary marker, but every such mark must be resolved before the proof is final.
Good proof-writing is about honest accounting. Every line is a transaction; every transaction has a reason; a proof that balances has no unexplained lines. Once you internalise this, the difference between your drafts and polished proofs shrinks dramatically.
Related: Mathematical Proof — Direct Proof · Every Step Justified — Direct Proof Template · How Formal Does My Proof Need to Be? · State Every Definition or Assume Reader Knows?