Write the proof in order
A proof has four moves. The order matters.
Step 1 of 4
- Suppose — assume √5 = p/q, in lowest terms.
- Square — this turns the guess into p² = 5q².
- Lemma twice — once for p, once for q.
- Contradiction — 5 divides both, which lowest terms forbids.
The rule
Each move uses the one before it. Squaring needs a guess to square. The second lemma needs q². The substitution is what gives q².
Extra working is allowed. A missing move breaks the proof.
The lemma is for a prime
Try the steps on √4. Its value is 2, and 2/1 squares to 4. The guess is true, so no contradiction appears. 4 is not prime.
The lemma here is for a prime.
Try them on √6. 6 is not prime. The lemma taught here does not cover 6. So these steps do not decide √6.
Watch out!
Skipping lowest terms breaks the last step. Without lowest terms, 4/8 has two even parts. Nothing is forbidden.
Now try it. Pick n. See where the steps work and where they stop.
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