$(\to)$ Assume that $S \subseteq R$ is bounded. If $(x_n)$ is a sequence in $S$ , then $(x_n)$ is bounded and thus it has a convergent subsequence.
$(\leftarrow)$ Assume that $S$ is not bounded. Then for any integer $n$ there is an $x_n$ in $S$ such that $|x_n| > n$. Since any convergent sequence is bounded, the sequence $(x_n)$ cannot have a convergent subsequence.
Can someone check to see whether my proof is missing something.