08. Upper and Lower Bounds

Upper Bound

Let $S\subseteq \mathbb{R}$ and $b\in \mathbb{R}$

$$b\text{ is an upper bound of }S\text{ if }s\le b\text{ for all }s\in S$$

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Lower Bound

Let $S\subseteq \mathbb{R}$ and $b\in \mathbb{R}$

$$b\text{ is an lower bound of }S\text{ if }s\ge b\text{ for all }s\in S$$

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Bounded Above

Let $S\subseteq \mathbb{R}$

$$S\text{ is bounded above }⟺S\text{ has an upper bound}$$

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Bounded Below

Let $S\subseteq \mathbb{R}$

$$S\text{ is bounded below}⟺S\text{ has a lower bound}$$

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Bounded

Let $S\subseteq \mathbb{R}$

$$S\text{ is bounded}⟺S\text{ is bounded above and below}$$

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