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| The \emph{infimum} of a set $S$ is the greatest lower bound of $S$ and is denoted $\inf(S)$. |
The infimum of a set $S$ is the greatest lower bound of $S$ and is denoted $\inf(S)$. |
| Let $A$ be a set with a partial order $\leq$, and let $S \subseteq A$. For any $x \in A$, $x$ is a lower bound of $S$ if $x \leq y$ for any $y \in S$. The infimum of $S$, denothed $\inf(S)$, is the greatest such lower bound; that is, if $b$ is a lower bound of $S$, then $b \leq \inf(S)$. |
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Let $A$ be a set with an order $\leq$, and let $S \subseteq A$. For any $x \in A$, $x$ is a lower bound of $S$ if $x \leq y$ for any $y \in S$. $\inf(S)$ is the greatest such lower bound; that is, if $b$ is a lower bound of $S$, then $b \leq \inf(S)$. |
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| Note that it is not necessarily the case that $\inf(S) \in S$. Suppose $S = (0, 1)$; then $\inf(S) = 0$, but $0 \not\in S$. |
Note that it is not necessarily the case that $\inf(S) \in S$. Suppose $S = (0, 1)$; then $\inf(S) = 0$, but $0 \not\in S$. |
| Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples. |
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