Recipe

Write a direct proof of a universal claim

Use when: the claim has the form "for every x in some set, P(x) implies Q(x)" (or just "for every x, Q(x)") and a forward chain from hypothesis to conclusion is at least plausible

Goal: an unbroken chain of justified steps from an arbitrary element satisfying the hypothesis to the conclusion, valid for every such element at once

Taught in Reading and writing a proof

1. 1

   Open with "let x be an arbitrary element of S" (or, when there's a hypothesis P(x), "let x be arbitrary with P(x)"). Whatever gets derived next must never depend on x being any particular value.

   Example

   For the claim \forall a,b\in\mathbb R,\ (a+b)^2\le2(a^2+b^2): "fix arbitrary real numbers a and b" — not, say, a=5 and b=-1.

   Why: the entire payoff of a direct proof — that it covers every element of S at once — depends on x never having been specialized; substituting a number here only proves the claim for that one number.

2. 2

   Restate the goal in a form where the property you need becomes visible: for an inequality, often as a difference (show it's \ge0); for an equation, by substituting a definition; for a set claim, by unfolding what membership means.

   Example

   Rewrite the inequality as the difference of its two sides and expand: 2(a^2+b^2)-(a+b)^2 = 2a^2+2b^2-a^2-2ab-b^2 = a^2-2ab+b^2

   Why: the claim in its original form rarely reveals why it's true; restating the goal as "show this expression is \ge0" turns it into something concretely checkable.

3. 3

   Write each line as a consequence of the line before it, and name the definition, identity, or already-established fact that licenses the step.

   Example

   Recognize a^2-2ab+b^2 as the perfect square (a-b)^2 — an algebraic identity, not a guess — and note (a-b)^2\ge0 because a real number squared is never negative.

   • the chain reaches the target statement directly — the proof is done; move to Step 4 and write the conclusion.
   • the chain stalls, with no obvious next justified step — go back to Step 2 and try a different rewriting of the claim.

   Why: every line needs a stated reason precisely so a reader can check each link, instead of taking the chain on faith.

4. 4

   Confirm the final line is exactly the conclusion you set out to prove, phrased in terms of the same arbitrary object from Step 1 — then say explicitly that, because that object was arbitrary, the result holds for every element of S.

   Example

   (a-b)^2\ge0 rearranges to (a+b)^2\le2(a^2+b^2); because a and b were arbitrary real numbers, this holds for every pair of real numbers.

   Why: this is the actual logical move that upgrades "true for this a,b" into "true for all a,b" — skipping it leaves even flawless algebra short of being a proof.

5. 5

   Substitute one concrete choice satisfying the hypothesis and confirm the conclusion holds numerically.

   Example

   At a=5,\ b=-1: left side (4)^2=16, right side 2(25+1)=52, and 16\le52 ✓.

   Why: this catches an algebra slip from Steps 2–3 before you trust the argument, but one example — or a thousand — never proves a claim about infinitely many elements; that's exactly what Steps 1–4 were for.