Subsets of intervals If S $\subseteq$ $\mathbb{R}$ is a nonempty, bounded set, and I := [inf S, sup S], show that S $\subseteq$ I.  Moreover, if J is any closed bounded interval containing S, show that I $\subseteq$ J.  To show that S $\subseteq$ I, let x $\in$ S.  Since S is bounded, inf S $\lt$ x < sup S.  Thus x $\in$ I.  Therefore S $\subseteq$ I.  Is this correct?  I feel like I am missing something.  How do I show that I $\subseteq$ J if S $\subseteq$ J? 

It's not true that $\inf S < x$; for example, if $S = [0, 1]$ and $x = 0$, this is false. What is true, and follows directly from the definitions, is that
$$\inf S \le x \le \sup S$$
since $\inf$ and $\sup$ are actually bounds on $S$. The result follows immediately from this observation. 

For your second question: Suppose $S \subseteq J = [a,b]$. Argue that $a \le \inf S$, so that $a$ is a lower bound for $I$. Make the same argument for the upper bounds.
