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.

Disproof

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.