proving inequalities — HSC mathematics extension 2, live

Proving Inequalities

Proving A ≥ B never requires knowing A or B exactly: write A − B as something that cannot be negative and the inequality falls out. For vectors, ‖u‖²‖v‖² − (u·v)² is the squared area of the parallelogram on u and v, which gives Cauchy–Schwarz, and Cauchy–Schwarz in turn bounds ‖u+v‖² to give the triangle inequality. The same move drives proof by cases, contrapositive and calculus arguments: find the gap, then show it is a square, a length or a function that never dips below zero. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

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Proving inequalities works like stacking evidence.

Proof isn't calculation — it's argument, where every line has to be forced by the one before it. Whether you're climbing an induction ladder or squeezing an inequality between what you know, you're never showing that something's true; you're showing it can't be false. The claim is the destination, but the reasoning is the whole mark.

Loose true facts sit in a pile — nothing proven yet, just raw material.

You lay each step on the last, every layer justified by the one below.

The stack reaches the top — the inequality now stands fully supported.

what examiners catch — Students write the chain backwards, starting from the statement to be proved and ending at a true line, or divide by an expression that may be negative without reversing the sign. What is rewarded is a forward chain that starts at a quantity known to be non-negative, says why it is non-negative, and justifies every square root or division.

five things, not forty.

Every inequality proof reduces to a gap you can show is non-negative: a perfect square, a squared area, a length or a function whose minimum is zero.

Cauchy–Schwarz: ‖u‖²‖v‖² − (u·v)² = ‖u‖²‖v‖² sin²θ is the squared area of the parallelogram on u and v, so it is never negative and |u·v| ≤ ‖u‖‖v‖.

Triangle inequality: ‖u+v‖² = ‖u‖² + 2u·v + ‖v‖² ≤ (‖u‖+‖v‖)² by Cauchy–Schwarz, and the number form |a+b| ≤ |a|+|b| follows by cases on the sign of b.

Direct proof is a forward chain of implications; proof by cases covers every sign; contrapositive and contradiction handle the equality case, where zero gap means parallel vectors.

Graph or calculus: let f(x) be one side minus the other, use f′ to locate its minimum and show f ≥ 0, as in eˣ ≥ 1 + x.

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 5 underneath this one, 0 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptproving inequalitiesTwo vectors with adjustable angle and length.

sits under itlogical connectives and quantifiersbecause an inequality proof is a statement to be established, so the language of implication comes first

sits under itplane vectors as arrows and componentsbecause The picture is built from u, v, their angle and their dot product.

sits under itscalar product and applications in three dimensionsbecause u·v = ‖u‖‖v‖cos θ is the identity the whole proof turns on.

sits under itdirect proof and proof by casesbecause The chain of implications and the sign cases are the techniques being applied.

sits under itproof by contradiction and counterexamplebecause The equality case is argued by contrapositive and contradiction.

they don't read it. they run it.

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Proving inequalities works like stacking evidence.

five things, not forty.

nothing here is a standalone fact.

3 more, same treatment.

they don't read it. they run it.