46 lines
2.2 KiB
TeX
46 lines
2.2 KiB
TeX
\section{The Completeness Property of $\R$}
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\begin{definition}
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Let $S$ be a nonempty subset of $\R$.
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\begin{enumerate}
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\item The set $S$ is said to be \textbf{bounded above} if there exists a number $u \in \R$ such that $s \leq u$ for all $s \in S$. Each such number $u$ is called an \textbf{upper bound} of $S$.
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\item The set $S$ is said to be \textbf{bounded below} if there exists a number $w \in \R$ such that $w \leq s$ for all $s \in S$. Each such number $w$ is called a \textbf{lower bound} of $S$.
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\item A set is said to be \textbf{bounded} if it is both bounded above and bounded below. A set is said to be \textbf{unbounded} if it is not bounded.
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\end{enumerate}
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\end{definition}
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\begin{definition}
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Let $S$ be a nonempty subset of $\R$.
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\begin{enumerate}
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\item If $S$ is bounded above, then a number $u$ is said to be a \textbf{supremum} (or a \textbf{least upper bound}) of $S$ if it satisfies the conditions:
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\begin{enumerate}
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\item $u$ is an upper bound of $S$, and
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\item if $v$ is any upper bound of $S$, then $u \leq v$.
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\end{enumerate}
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\item If $S$ is bounded below, then a number $w$ is said to be an \textbf{infimum} (or a \textbf{greatest lower bound}) of $S$ if it satisfies the conditions:
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\begin{enumerate}
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\item $w$ is a lower bound of $S$, and
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\item if $t$ is any lower bound of $S$, then $t \leq w$.
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\end{enumerate}
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\end{enumerate}
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\end{definition}
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\begin{lemma}
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A number $u$ is the supremum of a nonempty subset $S$ of $\R$ if and only if $u$ satisfies the conditions:
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\begin{enumerate}
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\item $s \leq u$ for all $s \in S$,
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\item if $v < u$, then there exists $s' \in S$ such that $v < s'$.
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\end{enumerate}
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\end{lemma}
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\begin{lemma}
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An upper bound $u$ of a nonempty set $S$ in $\R$ is the supremum of $S$ if and only if for every $\varepsilon > 0$ there exists an $s_\varepsilon \in S$ such that $u - \varepsilon < s_\varepsilon$.
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\end{lemma}
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\begin{theorem}[\textbf{The Completeness Property of $\R$}]
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Every nonempty set of real numbers that has an upper bound also has a supremum in $\R$. (This property is also called the \textbf{Supremum Property of $\R$}).
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\end{theorem}
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