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Explain Supremum

Date: 02/02/98 at 21:03:56
From: Jim Caprio
Subject: Explain supremum

Here's one that reveals my true identity as a non-mathematician - 
can you please explain, perhaps with an example, the concept of 
"supremum"?  Intuitively I want to think of it as "maximum," but it's 
not quite that simple, is it?

Jim Caprio

Date: 02/02/98 at 21:52:29
From: Doctor Pete
Subject: Re: Explain supremum


The concept of supremum, or least upper bound, is as follows:  Let

     S = {a[n]},

the sequence with terms a[0], a[1], ... over all the nonnegative 
integers. S has a supremum, called sup S, if for every n, 
a[n] <= sup S (i.e. no a[n] exceeds sup S), and furthermore, 
sup S is the *least* value with this property; that is, if 
a[n] <= b for all n, then sup S <= b for all such b.  

This is why the supremum is also called the "least upper bound," for a 
bound is a number which a function, sequence, or set, never exceeds.

Similarly, one can define the infimum inf S, or greatest lower bound.

Here are some examples illustrating sup and inf:  Let

     S = {a[n]}, a[n] = 1/n.

where n is a positive integer. Then sup S = 1, since 1/n > 1/(n+1) for 
all such n, and so the largest term is the first. However, inf S = 0.  
Notice that inf S is not an element in S! The reason why inf S = 0 is 
because if inf S = e > 0, there exists an integer N such that Ne > 1, 
by the Archimedean property of the reals. Hence e > 1/N, but 1/N is in 
S, hence inf S = 0 (since clearly 0 < 1/n).

-Doctor Pete,  The Math Forum
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