Prove or disprove if a+b=2 then b+z=3 for some z

Just a joke proof. Prove or disprove: for any two real numbers a and b such that a+b=2, then b+z=3 for some integer z.


Suppose that the statement is correct. Let a=0.5 and b=1.5. Then a+b=0.5+1.5=2 so that a and b satisfy the hypothesis of the statement. By the statement, there exists an integer z such that b+z=3. Then 1.5+z=3 by substitution. Subtracting 1.5 from both sides: z=1.5. But z=1.5 is not an integer. Thus, z is both an integer and not an integer, a contradiction. The hypothesis must be false and the statement is false, which was to be shown.