# How to convince a math teacher of this simple and obvious fact?

I have in my presence a mathematics teacher, who asserts that

Implies:

She has been shown in multiple ways why this is not true:

For me, these seem like valid (dis)proofs by contradiction, but she isn’t satisfied. She wants a ‘more mathematical’ proof, and I can’t think of any.

I’m worried that if she isn’t convinced, it may be detrimental to some students. Is there another way to systematically demonstrate the untruth of her conjecture?

EDIT: Since the answer which worked was from a comment, but each answer is also very good, I’m upvoting all of them instead of accepting a specific one. Feel free to close this question for being too open if so a moderator desires.

You can prove that all the numbers are equal 😉

Let’s assume that for all $a,b,c,d \in \mathbb{R}$, $b \neq 0$, $d \neq 0$ we have

Now take any two numbers, say $p$ and $q$, and write

Using claim $(\spadesuit)$ we have $p = q$. For the special case, where one of them equals zero (e.g. $q$), use

I hope this helps 😉