Explain SupremumDate: 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 Hi, 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 Check out our web site! http://mathforum.org/dr.math/ |
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