Unbounded Space Minus Bounded Space is Unbounded/Proof 2

Proof
$A \setminus B$ is bounded.

By Finite Union of Bounded Subsets:
 * $\paren {A \setminus B} \cup B$

is bounded.

On the other hand, by :
 * $A \subseteq \paren {A \setminus B} \cup B$

Thus by Subset of Bounded Subset of Metric Space is Bounded, $A$ must be bounded, too.

This is a contradiction.