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Definition (Infimum And Supremum)

When a set $S$ is bounded below, there are infinitely many lower bounds. The infimum of $S$ is the greatest lower bound, which we write as $\inf S$. So if $S$ is a set, then we write $m=\inf S$ if and only if both

$m$ is a lower bound for $S$, and
$m$ is the greatest lower bound for $S$ (if $m'$ is another lower bound, then $m\geq m'$).

When a set $S$ is bounded above, there are infinitely many upper bounds. The supremum of $S$ is the least upper bound, which we write as $\sup S$. So if $S$ is a set, then we write $m=\sup S$ if and only if both

$m$ is an upper bound for $S$, and
$m$ is the least upper bound for $S$ (if $m'$ is another upper bound, then $m\leq m'$).

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Page last modified on March 11, 2015, at 12:48 PM

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HomePage Schedule Pictures Writing Tips LaTeX Commands External Resources List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Definition Infimum And Supremum Please Login to access more options. Definition (Infimum And Supremum) When a set $S$ is bounded below, there are infinitely many lower bounds. The infimum of $S$ is the greatest lower bound, which we write as $\inf S$. So if $S$ is a set, then we write $m=\inf S$ if and only if both $m$ is a lower bound for $S$, and $m$ is the greatest lower bound for $S$ (if $m'$ is another lower bound, then $m\geq m'$). When a set $S$ is bounded above, there are infinitely many upper bounds. The supremum of $S$ is the least upper bound, which we write as $\sup S$. So if $S$ is a set, then we write $m=\sup S$ if and only if both $m$ is an upper bound for $S$, and $m$ is the least upper bound for $S$ (if $m'$ is another upper bound, then $m\leq m'$). Edit Page History Source Attach File Backlinks List Group Page last modified on March 11, 2015, at 12:48 PM
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