[Def]: Bounded Above and Bounded Below

Given $S\subseteq \R$, $S\neq \phi$

1. If $\exists U$ such that $s\leq U,\forall s\in S$, then $S$ is said to be bounded above, and $U$ is an upper bound of $S$
2. If $\exists l$ such that $s\geq l,\forall s\in S$, then $S$ is said to be bounded below, and $l$ is a lower bound of $S$

[Def]: Supremum and Infimum

Given $S\subseteq \R$, $S\neq \phi$

1. If $S$ is bounded above, define $\sup S$ is the smallest upper bound of $S$. Else (not bounded above) define $\sup S=+\infty$
2. If $S$ is bounded below, define $\inf S$ is the greatest lower bound of $S$. Else (not bounded below) define $\inf S=-\infty$

性質: Let $w=\sup S$

1. $s\leq w, \forall s\in S$
2. $\forall \varepsilon>0,\exists s_\varepsilon\in S$ such that $s_\varepsilon>w-\varepsilon$