Can someone help me prove or disprove

I need to Prove or Disprove: There does not exist a real number x such that for all real numbers y, xy=1.

This is false. I am going to try to prove it using contradiction.

Proof:

Suppose not. That is, suppose that there **does** exist a real number x such that for all real numbers y, xy=1. Let x be any real number r. X=1/r. Let y be an real number r. Y=r/1

Then,

X*Y= (1/r)*(r/1) by substitution

= 1 By algebra.

This is a contradiction because there **does** exist a real number x such that for all real numbers y x*y=1. **end of proof**

Can some someone please let me know if this is a correct way to write it on paper? I am pretty confident, however I would like to make sure, is there anything I can add for correctness?

Thank You,

Matt H.