Error in demonstration of $(a-b)^{2} \ge 0$

Question

It's a fact that:

$\frac{a+b}{2} \ge \sqrt{ab}$

When a and b are non negative real numbers. What's the error in the following demonstration:

$\frac{a+b}{2} \ge \sqrt{ab}$ , then

$a+b \ge 2\sqrt{ab}$ , then

$a^{2}+2ab+b^{2} \ge 4ab$ , then

$a^{2}-2ab+b^{2} \ge 0$ , then

$(a-b)^{2} \ge 0$

discrete math professor gave that question as homework, and this is a direct translation from portuguese.

the only idea I have is to say that both $(a-b)^{2}\ge 0$ and $(b-a)²\ge 0$ are valid, but it only specifies one. Is that correct?

Welcome to MSE. Who says there’s an error? $(a-b)^2=(b-a)^2$ — J. W. Tanner, Commented Mar 18 at 0:03
How do you "know" $(a+b)/2 > \ge \sqrt{ab}$? The usual way to prove that starts with $(a-b)^2 \ge 0$. — Ethan Bolker, Commented Mar 18 at 0:07
Possible duplicates: math.stackexchange.com/q/595034/532409 math.stackexchange.com/q/2343337/532409 math.stackexchange.com/q/557653/532409 math.stackexchange.com/q/4614921/532409 — Quillo, Commented Mar 18 at 0:18
There's nothing wrong with that demonstration. It just doesn't mean anything as $(a-b)^2\ge 0$ is always true =whether $\frac {a+b}2 \ge \sqrt ab$ or not. You might as well say "Fish like tacos $\implies a-b\in \mathbb R \implies (a-b)^2 \ge 0$. The thing though is EVERY step is reversible. $\frac {a+b}2\ge \sqrt{ab}\iff a+b\ge 2\sqrt{ab}\iff (a+b)^2\ge 4ab$ AND $a,b\ge 0\iff a^2+2ab +b^2 \ge 4ab;a,b\ge 0\iff a^2-2ab-b^2\ge 0;a,b\ge 0\iff (a-b)^2\ge 0;a,b\ge 0$. As $(a-b)^2\ge 0$ always and we were told $a,b\ge 0$ this means we have proven $\frac {a+b}2\ge \sqrt{ab}$. — fleablood, Commented Mar 18 at 0:32

Starlight · Accepted Answer · 2024-03-18 06:31:34Z

It's a common mistake in proofs to assume the truth of what you wish to prove. Once that has been done, you can prove $anything$.

For example, here is an absurd "proof" that $$2=3\Rightarrow 0=0\Rightarrow \text{Last statement is valid}$$

The above proof would have been correct if you begin with the last step, and work backwards.

Or, if you write at the very end that: All steps are reversible.

That last statement, of course, needs to be checked very carefully.

1 Answer 1