What is wrong with this proof that 3 is less than 1?

Question

What is wrong with this proof?
Theorem. 3 is less than 1.
Proof. Every number is either less than 1 or greater than 1 or equals 1. Let $c$ be an arbitrary number. Therefore, it is less than 1 or greater than 1 or equals 1. Suppose it is less than 1. By the rule of universal generalization, if an arbitrary number is less than 1, every number is less than 1. Therefore, 3 is less than 1.

Answer 1

You are fine up to "Suppose it is less than 1" You need to review your rule of universal generalization (which you did not quote), which will likely fail here.

Answer 2

Once you've decided $c$ is less than $1$, it's not arbitrary anymore.