AM-GM-HM inequalities — proofs: Foundations

Everything below is a prerequisite for the AM–GM–HM proofs. We start from nothing.

0. What is a "positive real number"?

Why does the whole topic insist on this? Because we are about to take square roots and reciprocals ( $1/ a$ ). A square root of a negative number is not a point on our line, and $1/0$ does not exist. So the ticket to enter this topic is: all numbers strictly positive. Every proof quietly leans on this.

1. The two basic verbs: $+$ and its undo $-$

We call this out early because the master trick of the whole topic (Section 8) starts from a subtraction, $a - b$ . The topic needs a solid grip on " $-$ undoes $+$ " before that step makes sense.

2. The other basic verb: $\times$ , its undo $\div$ , and powers $x^{n}$

3. Subscripts: naming many numbers at once — $a_{1}, a_{2}, \dots, a_{n}$

$a_{1}$ = the first number, $a_{2}$ = the second, $a_{n}$ = the last.
$n$ = how many numbers there are in total.
$a_{i}$ = "the $i$ -th one, whichever that is."

The topic needs this because AM, GM, HM are all statements about a whole list $a_{1}, \dots, a_{n}$ . Without subscripts you could only ever talk about two or three specific numbers.

4. The summation sign $\sum$ — "add them all up"

The $\sum$ (Greek capital S, for Sum) is just a machine that runs the counter and adds. The bottom ( $i = 1$ ) says where to start, the top ( $n$ ) says where to stop. Since $n \geq 1$ (Section 3), the machine always runs at least once — we never meet an "empty sum" here.

This is the raw material of the Arithmetic Mean: sum everything, then share it out. It also directly connects to Arithmetic Progressions, where you first met sums of listed terms.

5. The product sign $\prod$ — "multiply them all together"

$\prod$ is Greek capital P (for Product) — the multiplication twin of $\sum$ . Same counter, same start/stop, but the operation is $\times$ not $+$ . Again $n \geq 1$ , so there is always at least one factor — no "empty product" to puzzle over.

This is the raw material of the Geometric Mean, and it is the natural language of Geometric Progressions, where terms grow by multiplying.

6. Roots and fractional powers — and $x^{1/ n}$

The GM formula $(a_{1} \dots a_{n})^{1/ n}$ is literally: take the product, then take the $n$ -th root. And notice — the root needs its input to be positive (Section 0), or we'd be asking for the square root of a negative.

7. The reciprocal — $\frac{1}{a}$ — and why it "flips" the world

The HM is built by: flip every number, average the flips, then flip back. That's why HM lives in the world of Harmonic Progression and rate problems (speed, work, resistance) — anything measured per unit of something.

8. The inequality signs $\geq$ , $\leq$ , and "iff"

The whole topic is an inequality: $AM \geq GM \geq HM$ . The word iff shows up in the equality condition ("equality iff all numbers are equal") — meaning the means coincide exactly when the numbers are all identical, and only then.

9. The master square: $(\cdot)^{2} \geq 0$

Formula Expanding the master square — the missing step in full

First, why does $(p - q)^{2} = p^{2} - 2 pq + q^{2}$ ? Because $(p - q)^{2}$ means $(p - q) \times (p - q)$ , and multiplying out — each term in the first bracket times each in the second — gives $p \times p - p \times q - q \times p + q \times q = p^{2} - 2 pq + q^{2}$ (the two identical middle terms combine into $- 2 pq$ ; the " $-$ " here is subtraction from Section 1). Now put $p = a, q = b$ into the always-true fact: $(a - b)^{2} \geq 0 ⟹ = a (a)^{2} - 2 = ab a b + = b (b)^{2} \geq 0.$ Why each piece: $(a)^{2} = a$ and $(b)^{2} = b$ because squaring undoes the square root (Section 6), and $a b = ab$ by the "roots multiply" identity just proved. So the left side is $a - 2 ab + b$ . Now move the middle term to the other side (add $2 ab$ to both sides — using that $-$ undoes $+$ , Section 1): $a + b \geq 2 ab .$ This is $AM \geq GM$ for two numbers (divide both sides by $2$ ). The $= 0$ case ( $a = b$ , i.e. $a = b$ ) is precisely the equality condition. One tiny fact, the whole chain.

How the foundations feed the topic

Once these are solid you are ready for Weighted Means (Power Mean Inequality), Cauchy-Schwarz Inequality, Jensen's Inequality, and Optimization using inequalities — all of which reuse exactly these bricks.

Equipment checklist

Cover the right-hand side and test yourself.

What does

a > 0

guarantee we're allowed to do?

Take square roots and reciprocals safely — no undefined operations.

What operation undoes addition, and what undoes multiplication?

Subtraction undoes addition; division undoes multiplication.

What does the exponent in

x^{n}

count?

The number of copies of

x

multiplied together:

x^{n} = x \times x \times \dots \times x

(

n

times).

What precisely does the ellipsis

\dots

a_{1}, \dots, a_{n}

mean?

Continue the obvious pattern one step at a time, every box present, up to the stated last term

a_{n}

— not "forever".

What kind of numbers are the counter

i

and total

n

, and what is ruled out?

Positive integers (

1, 2, 3, \dots

); the empty list

n = 0

is excluded — nothing to average.

What does the subscript

a_{i}

let one formula do?

Talk about a whole list of numbers at once, using a counter

i

Expand

i = 1 \sum n a_{i}

in words.

a_{1} + a_{2} + \dots + a_{n}

— add every term from the first to the last.

Expand

i = 1 \prod n a_{i}

in words.

a_{1} \times a_{2} \times \dots \times a_{n}

— multiply every term.

What question does

x^{1/ n}

answer, and which operation does it undo?

Which positive number multiplied by itself

n

times gives

x

? — it undoes the power

x^{n}

Why is

a b = ab

for positive

a, b

Squaring

a b

gives

ab

, and it is positive, so it must be

ab

On the reciprocal seesaw, where does

2

land?

\frac{1}{2}

— above

1

flips to below

1

a \leq b

(both positive), what happens to

\frac{1}{a}

\frac{1}{b}

The inequality flips:

\frac{1}{a} \geq \frac{1}{b}

How does

(p - q)^{2}

expand, and why?

p^{2} - 2 pq + q^{2}

, from multiplying

(p - q) \times (p - q)

term by term.

Expand

(a - b)^{2} \geq 0

— what inequality pops out?

a - 2 ab + b \geq 0

, i.e.

a + b \geq 2 ab

(that's AM

\geq

GM).

What does "AM = GM iff all equal" claim in both directions?

All numbers equal forces the means equal, AND the means equal forces the numbers equal.

Which single fact powers every

\geq

in the proofs?

A square is never negative:

t^{2} \geq 0

, with equality only at

t = 0