Topology: An Inquiry-Based Approach

A lower bound or a nonempty subset $S$ of $\mathbb{R}$ that is bounded below is a real number $m$ such that $m\le s$ for all $s\in S\text{.}$ A greatest lower bound (or infimum) for a nonempty subset $S$ of $\mathbb{R}$ that is bounded below is a real number $m$ such that

An upper bound for a nonempty subset $S$ of $\mathbb{R}$ that is bounded above is a real number $M$ such that $M\ge s$ for all $s\in S\text{.}$ A least upper bound (or supremum) for a nonempty subset $S$ of $\mathbb{R}$ that is bounded above is a real number $M$ such that