If $\displaystyle S = \{ x \in \mathbb{R}: x(x-3) \leq 0 \} $ and $\displaystyle T = \{x \in \mathbb{R}: x \geq 3 \} $, then every element of $\displaystyle T $ is an upper bound for the set $\displaystyle S $.

So $\displaystyle S = [0,3] $ which is a compact set, and so closed and bounded. Consequently there exists a least upper bound, and $\displaystyle \sup S = 3 $ and so $\displaystyle T $ is a set of upper bounds for $\displaystyle S $.

Is this correct?