Let $\mathbb G(q,l)$ to be the set of all floating-point numbers in the base $q$ and with a mantissa of a length $l$. In the lecture notes I am trying to learn from, the following is said to be obvious (and consequently has no proof):
$$ \forall x \in \mathbb G(q,l), \forall y \in \mathbb G(q,l)\setminus\{x\}: |x-y| \ge |x|q^{-l} $$
What I have tried so far:
Let $x=(-1)^m a \cdot q^\alpha$ and $y=(-1)^n b \cdot q^\beta \in \mathbb G(q,l)$ such that $x \ne y$, then
\begin{align} |x-y| &\ge |x|-|y| \\ &=|(-1)^m a \cdot q^\alpha|-|(-1)^n b \cdot q^\beta| \\ &=a \cdot q^\alpha - b \cdot q^\beta \\ &\ge ...? \end{align}