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**matthayzon89** Prove or disprove: There does not exist a real number x such that for all real numbers y, xy=1.

Contradiction: Suppose yes. That is, suppose that there is a real number x such that for all real numbers y, x*y=1.

Let x be the real number 0 and let y=z for any real number z.

Then,

x*y= 0*z .........by substitution

= 0 ......... by algebra

There is a contradiction, because when x is the real number 0, x multiplied by y equals 0. Therefore, the original statement is in fact true. There does not exist a real number x such that for all real numbers y, xy=1. **end of proof**

The only concern w/ my proof: Isn't the statement a universal statement? I thought that you are not allowed to use numbers to PROVE a universal statement using actual numbers, but this is the only proof I was able to come up with. Also, isn't this prove more of a counterexample as opposed to a contradiction?