Prove that this is no real number such that $x \leq a$ for all real $x$.
I want to know if the way I proved it is valid or not.
Proof. We first prove that there is no real number $a$ such that $x=a$ for all real $x$. If this were true, then all real numbers would be equal to $a$. This is not possible, because the axiom that garantees the existence of identity elements tells us that the set of real numbers has the numbers $0$ and $1$. Therefore, the set of real numbers has more than only one element.
Is this proof correct? Can I improve it? Thank you.