Here is a claim: for any two positive real numbers a and b, the quantity (a + b)/2 is never less than √(ab). You might verify this for a few examples. You might even believe it intuitively. But how do you prove it—with absolute certainty, for every possible pair of positive reals?

Inequality proofs are among the most satisfying constructions in mathematics. Unlike equations, which assert balance, inequalities assert dominance—one side always wins. Proving that requires a different toolkit than solving for x. You cannot simply rearrange and isolate. You need strategies: architectural plans for building an argument that leaves no room for doubt.

This article develops three foundational strategies for proving inequalities. Each one is a lens through which entire families of problems become transparent. Together, they form a proof architect's essential toolkit—one that transforms the question "Is this true?" into the far more powerful question "Why must this be true?"

The most counterintuitive strategy in inequality proofs is also one of the most powerful: start from something you already know is true, and work toward what you want to prove. This feels backward. Shouldn't a proof move from hypothesis to conclusion? It can. But for inequalities, the reverse direction often reveals the path.

Consider proving that for positive reals a and b, we have a² + b² ≥ 2ab. Rather than staring at this cold, start from an unassailable truth: (a − b)² ≥ 0. This is true because a square is never negative. Now expand: a² − 2ab + b² ≥ 0. Rearrange: a² + b² ≥ 2ab. Done. Each step is reversible—every transformation preserves the inequality's direction. The proof is logically airtight in both directions.

This technique is sometimes called working backward or analysis-synthesis. In the analysis phase, you assume the result and deduce what simpler truth it must rest on. In the synthesis phase, you reverse the chain, presenting the argument from known truth to desired conclusion. What you publish is the synthesis. What you discover is the analysis.

The key requirement is reversibility. Every step must be an equivalence or an implication that works in the direction you need. Adding the same quantity to both sides, multiplying both sides by a positive number, squaring both sides when both are non-negative—these are safe. Squaring when signs are unknown, or dividing by a variable that might be zero, can break the chain. The logical architect checks every link.

Takeaway

When an inequality seems opaque, try assuming it's true and simplifying until you reach something obviously true. Then reverse the steps. Discovery and presentation are two different journeys through the same logical landscape.

One of the most versatile tools in the inequality prover's arsenal is the AM-GM inequality: for non-negative reals, the arithmetic mean is always at least as large as the geometric mean. For two numbers: (a + b)/2 ≥ √(ab). For n numbers: (a₁ + a₂ + … + aₙ)/n ≥ (a₁ · a₂ · … · aₙ)^(1/n). Equality holds exactly when all the numbers are equal.

The two-variable proof is elegant and uses our first strategy. Start from (√a − √b)² ≥ 0. Expand: a − 2√(ab) + b ≥ 0. Rearrange: a + b ≥ 2√(ab). Divide by 2: (a + b)/2 ≥ √(ab). Four lines, total certainty. The n-variable case requires more machinery—Cauchy's forward-backward induction is a classic approach—but the core insight is the same: the gap between arithmetic and geometric means measures how "spread out" the values are.

AM-GM sits in a broader chain: HM ≤ GM ≤ AM ≤ QM, where HM is the harmonic mean (n / Σ(1/aᵢ)) and QM is the quadratic mean (√(Σaᵢ²/n)). Each inequality in this chain has its own proof and its own applications. The harmonic-geometric inequality, for instance, appears naturally in problems involving rates and reciprocals.

The power of AM-GM lies in its universality as a bounding tool. Need to show that x + 1/x ≥ 2 for positive x? Apply AM-GM to x and 1/x directly. Need to minimize a sum of terms whose product is fixed? AM-GM gives the answer: the minimum occurs when all terms are equal. Recognizing when a problem secretly asks about the relationship between a sum and a product is the skill that turns AM-GM from a theorem into a weapon.

Takeaway

AM-GM converts questions about sums into questions about products and vice versa. Whenever you see a sum you want to bound and a product you can control—or the reverse—the mean inequalities are likely your shortest path to a proof.

If there is a single atomic fact underlying most elementary inequality proofs, it is this: for any real number x, x² ≥ 0. This is not a deep theorem. It is almost too obvious to state. And yet an astonishing number of important inequalities are, at their core, just this fact dressed up in different clothing.

We already saw it power the proof of a² + b² ≥ 2ab. But the technique extends much further. The Cauchy-Schwarz inequality—one of the most important in all of mathematics—states that (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²). One clean proof proceeds by considering the expression Σ(aᵢt − bᵢ)² for a real parameter t. This is a sum of squares, so it is ≥ 0 for all t. Expanding gives a quadratic in t that is always non-negative, which means its discriminant must be ≤ 0. That discriminant condition is Cauchy-Schwarz.

The technique of completing the square is the algebraic embodiment of this principle. Any quadratic expression can be rewritten to expose a squared term plus a remainder. If you can show the remainder is also non-negative, the entire expression is non-negative. This is exactly how we prove that a quadratic with negative discriminant has no real roots—the completed-square form makes the positivity visible.

The deeper lesson is one of proof architecture: many inequality proofs reduce to finding the right expression to square. The challenge is not verifying that squares are non-negative—that is trivial. The challenge is constructing the right squared quantity. This is where creativity enters. You might need to introduce auxiliary variables, group terms cleverly, or apply substitutions that transform the problem into one where the squared structure becomes apparent. The fact itself is simple. The art is in the construction.

Takeaway

The non-negativity of squares is the bedrock axiom of inequality proofs. When you're stuck on an inequality, ask: is there a way to express the difference between the two sides as a sum of squares? If you find it, the proof writes itself.

Three strategies. One common thread: inequality proofs succeed when you find the right structure to exploit. Working backward reveals that structure. AM-GM provides it ready-made for sums and products. And the non-negativity of squares offers the most elemental building block of all.

These techniques are not isolated tricks. They interlock. AM-GM is proved using squares. Cauchy-Schwarz is proved using a discriminant argument that relies on squares. The backward-working strategy helps you discover which tool to apply. Mastering one sharpens your use of the others.

What makes inequality proofs deeply satisfying is their finality. Once constructed, the argument is unassailable. No counterexample can exist. No edge case was missed. The logical architecture guarantees it. That is the power of proof—not just knowing that something is true, but understanding why it must be.