How to show that a bounded subset of a metric space is contained within a closed sphere using the definition of a bounded set? Source: Introduction to Topology and Modern Analysis by GF Simmons(Pg-69, Q.No.3)
How to show that a bounded non-empty subset of a  metric space is contained within a closed sphere using the definition of a bounded set as given below?
A set is bounded if its diameter is finite (a real number less than infinity).
Also see that,
given a metric space $(X,d)$, the diameter of nonempty subset $ M \subseteq X$ is defined as follows:
$$\mathrm{diam}(M)=\displaystyle\sup_{x, y \in M} d(x, y)$$

Just pick a point in the given subset, $P\in A$ and consider the ball centered at $P$ with radius $R$ equal to the diameter of $A$. Then clearly for any $Q\in A$, $dist (Q,P)\le R$ by definition of diameter.
