2.3 The Completeness Property of R

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2.3.1 Definition: Bounded set

Let $S$ be a nonempty subset of $\mathbb{R}$ ($S\ne \varnothing ,S\subseteq \mathbb{R}$)

1. If $\exists u\in \mathbb{R}\ni s\le u,\forall s\in S$, then we say $S$ is bounded above
2. If $\exists u\in \mathbb{R}\ni s\ge u,\forall s\in S$, then we say $S$ is bounded below
3. $S$ is bounded if $S$ is bounded above and below

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2.3.2 Definition: Supremum and infimum

Let $S\subseteq \mathbb{R},S\le \varnothing$

1. If $S$ is bounded above, then a number $u$ is said to be a supremum (or a least upper bound) of $S$ if it satisfies the conditions:

   1. $u$ is an upper bound of $S$, and
   2. If $v$ is any upper bound of $S$, then $u\le v$.

2. If $S$ is bounded below, then a number $w$ is said to be an infimum (or a greatest lower bound) of $S$ if it satsifies the conditions:

   1. $w$ is a lower bound of $S$, and
   2. If $t$ is any lower bound of $S$, then $t\le w$.

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Proposition: Uniqueness of supremum

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2.3.3 Lemma: Formal definition of supremum and infimum

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2.3.4 Lemma: Limits in supremum and infimum

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2.3.6 Definition: Completeness property of R

Every nonempty set of real numbers that has an upper bound also has a supremum in $\mathbb{R}$.

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