# 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.

## 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.