Solution
= Solution
Write each member as $3-2/(n+1)$. Every member is strictly below $3$, so $3$ is an <upper bound>. If $u<3$, the <Archimedean property> supplies $n\in\mathbb N$ with $n+1>2/(3-u)$; then $3-2/(n+1)>u$. Thus no smaller number is an <upper bound>, proving \b[$\sup S=3$]. Since $3\notin S$, there is \b[no <greatest element>]. Equivalently, each member is followed by a larger member because $2/(n+2)<2/(n+1)$.