TITLE: Diameter of a set.
QUESTION [5 upvotes]: Suppose $A$ is a nonempty set in a metric space $(X, d)$. Define
$$ \delta(A) = \sup_{x,y \in A} d(x,y) $$
Is it true that if $A \subseteq B$, then $\delta(A) \leq \delta(B) $??

REPLY [3 votes]: Recall the definition of the supremum: It's the least upper bound on a set of real numbers. Now if $C \subseteq D$ are sets of real numbers, every upper bound on $D$ is an upper bound on $C$; this proves that $\sup C \le \sup D$.
Finally, since $A \subseteq B$, it's immediate that
$$\{d(x, y) : x, y \in A\} \subseteq \{d(x, y) : x, y \in B\}$$
and by applying the previous paragraph, we're done.