I am reading How to prove it and doing the exercise in session 3.3.
Theorem: For every real number $x$, $x^2 \geq 0$.
What's wrong with the following proof of the theorem?
Proof. Suppose not. Then for every real number $x$, $x^2 \lt 0$. In particular, plugging in $x = 3$ we would get $9 \lt 0$, which is clearly false. This contradiction shows that for every number $x$, $x^2 \geq 0$.
This proof makes sense to me but why it is incorrect?