Suppose that S is a nonempty bounded set of real numbers and T is a nonempty subset of S.
Show that T is bounded
The statement that $\displaystyle S$ is a bounded set of real numbers means that $\displaystyle \left( {\exists B > 0} \right)\left[ {\left( {\forall z \in S} \right) \Rightarrow \left| z \right| \leqslant B} \right]$.
Therefore $\displaystyle T \subseteq S\,\& \,x \in T \Rightarrow \quad \left| x \right| \leqslant B$.
That shows that $\displaystyle T$ is bounded.