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- Thread starter Bashyboy
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The supremum of a set A is the least upper bound and the infimum is the greatest lower bound.

For example, consider the set [0,1]. This has many upper bounds. For example, 2, 10, 10020330 are all upper bounds. But 1 is the smallest such upper bound. Thus 1 is the supremum.

In the previous example, 1 was actually a maximum: that is, the greatest element contained in the set. But a supremum does not need to be contained in the set. For example, ]0,1[ (or (0,1) in other notation) also has 1 as smallest upper bound. Every element smaller than 1 will not be an upper bound anymore. Thus 1 is the supremum of the set.

The same discussion holds for infima.

For example: [tex]inf ]0,2[ = 0[/tex] or somewhat more complicated [tex]inf \{1/n~\vert~n\in \mathbb{N}\}=0[/tex] (note that I take [itex]0\in \mathbb{N}[/itex]).

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So, are the words, in a way, synonymous to minimum and maximum?

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So, are the words, in a way, synonymous to minimum and maximum?

No!! That was the entire point. The supremum is a generalization of the maximum.

When I say that an element x is the maximum of A, then this means that x is the greatest element contained in A. So it is implied that x is an element of A.

But with the supremum, x does not need to be an element of A. For example sup ]0,1[ = 1, but 1 is not an element of ]0,1[.

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HallsofIvy

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For example, the open interval, (0, 1)= {x| 0< x< 1}, does not have a "minimum" or maximum. But its inf (also called "greatest lower bound") is 0 and its sup (also called "least upper bound") is 1.

**If** a set has a "minimum" then its infimum is that minimum. Similarly, if a a set has a "maximum" then its supremum is that maximum. But any set with a lower bound has an infimum but not necessarily a "minimum" and any set with an upper bound has a supremum but not necessarily a "maximum".

(Some texts allow "[itex]-\infty[/itex]" as an infimum and "[itex]\infty[/itex]" as a supremum so that a set does not have to be bounded to have infimum and supremum.)

(Some texts allow "[itex]-\infty[/itex]" as an infimum and "[itex]\infty[/itex]" as a supremum so that a set does not have to be bounded to have infimum and supremum.)

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