what is wrong with the following proof that 1 is the largest real number?

1 is the largest real number :

proof:

let x be the largest real number (want to prove that x=1) then

(since and x is the largest)

so that

hence, either

but is impossible so and therefore

or x(x-1)=0 so that x=0 or x=1 since 1>0 then x=1 and thus 1 is the largest real number