Absolute Value of Integer is not less than Divisors

Theorem
A (non-zero) integer is greater than or equal to its divisors in magnitude:


 * $\forall c \in \Z^*: a \mathop \backslash c \implies a \le \left\vert{a}\right\vert \le \left\vert{c}\right\vert$

It follows that a non-zero integer can have only a finite number of divisors, since they must all be less than or equal to it.

Corollary
Let $a, b \in \Z$.

If $a$ and $b$ are both positive, and $a \mathop \backslash b$, then $a \le b$.

Proof
Suppose $a \mathop \backslash c, c \ne 0$. It's a given that $a \le \left\vert{a}\right\vert$.

Proof of Corollary
Follows directly.