Union of two bounded sets is bounded Suppose $(X,d)$ is a metric space, for a nonempty subset $A$ of $X$, define
$$ \delta(A) = \sup_{x, y \in A} d(x,y) $$
$A$ is bounded if $\delta(A) < \infty $. If $A$, $B$ are bounded, does it follow that $A \cup B$ is bounded ? I know that $A \cap B$ is bounded since $A \cap B \subseteq B$, and the fact that $A \subseteq B \implies \delta(A) \leq \delta(B) $. But how about the $\cup$ ?
thanks

Let's $a\in A$ and $b\in B$ be fixed. Take any $x\in A , y\in B$, then $$d(x,y) \leq d(x,a ) +d(a,b) +d(b,y) \leq \delta (A) +d(a,b) +\delta (B)$$
thus $A\cup B$ is bounded.
