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

For all $S\subseteq \mathbb{R}$, $\sup S$ is unique.

In other words,

If $u_1=\sup S$ and $u_2=\sup S$, it must be that $u_1=u_2$.

IMPORTANT

• If $u^{\prime }$ is an arbitrary upper bound of $S$, then $\sup S\le u^{\prime }$.
• There are 4 possibilities for a nonempty subset $S$ of $\mathbb{R}$: it can
  1. Have both a supremum and an infimum.
  2. Have a supremum but no infimum.
  3. Have an infimum but no supremum.
  4. Have neither a supremum nor an infimum.
• In order to show that $u=\sup S$ for some nonemtpy subset $S$ of $\mathbb{R}$, we need to show (1) and (2) of 2.3.2 (1) holds. This can also be done using 2.3.3 Lemma or 2.3.4 Lemma.
• The following statements about an upper bound $u$ of a set $S$ are equivalent:
  1. If $v$ is any upper bound of $S$, then $u\le v$.
  2. If $z<u$, then $z$ is not an upper bound of $S$.
  3. If $z<u$, then there exists $s_z\in S$ such that $z<s_z$.
  4. If $\epsilon >0$, then there exists $s_{\epsilon }\in S$ such taht $u-\epsilon <s_{\epsilon }$.

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2.3.3 Lemma: Formal definition of supremum and infimum

Let $S$ be a nonempty subset of $\mathbb{R}$. $u=\sup S$, if and only if:

1. $s\le u,\forall s\in S$
2. If $v<u$ then there exists an $s^{\prime }\in S$ such that $v<s^{\prime }$

Point 2 is saying:

If $v$ not an upperbound of $S$ then there exists an element in $S$ that is bigger than $v$

NOTE

This is the negation of (2) of 2.3.2 (1). Here $v$ is under $u$, not above.

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2.3.4 Lemma: Limits in supremum and infimum

def-supremum-and-infimum_202508160109#lemma-limits-in-supremum-and-infimum

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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