This statement is false. As a counterexample, $3$ and $1$ are both odd, but $3-1=2$ and $2$ is even. Therefore, the difference of two odd integers is not necessarily odd.
Work Step by Step
The methods of this section allow us to show us the opposite: that an odd minus an odd is always even. In short, we have $(2m+1)-(2n+1)=2m+1-2n-1=2m-2n=2(m-n)$, where $2(m-n)$ is clearly even.