Upper Bound for Lowest Common Multiple

Theorem
Let $a, b \in \Z$ be integers such that $a b \ne 0$.

Then:
 * $\lcm \set {a, b} \le \size {a b}$

where:
 * $\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$

Proof
By Product of GCD and LCM:
 * $\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$

where:
 * $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.

By Existence of Greatest Common Divisor $\gcd \set {a, b}$ exists.

By definition of GCD, $\gcd \set {a, b} \in \Z_{>0}$.

Hence the result.