Is this proof correct?
Theorem: The product of an even integer and an odd integer is even.
Proof: Let $a$ and $b$ be integers. Assume $a$ is even and $b$ is odd, so there exists an integer $p$ so that $a=2p$ and there exists an integer $q$ so that $b=2q+1$. If $a \cdot b$ is even then by definition of even there exists an integer $r$ such that $a \cdot b = 2r$. So we have $a \cdot b = (2p) (2q+1) = 2r$, where $r$ is an integer.
Therefore, $a \cdot b$ is even.