I have that by definition an integer $n$ is even if $n = 2m$ for some integer $m$. By definition an integer is odd if it is not even. I would like to prove that if $n$ is odd, then $n = 2m + 1$ for some $m$.
I am supposed to show this using at little as possible about the properties of the integers. I don't, for example, know anything about division algorithms like Euclidean division.
I can show that all numbers of the form $2m + 1$ are odd, but how can I show that all integers that are odd are indeed of this form?