Re: sup, inf

David W. Cantrell <DWCantrell@xxxxxxxxxxx> wrote:
fjblurt@xxxxxxxxx wrote:
On Sep 1, 1:58 pm, ame <rael...@xxxxxxxxx> wrote:
how would one go about proving that sup(A/B) = sup A / inf B, with
A,B>= 0 ? i'm not sure that following the same argument i used for
sup(AB) = sup A sup B will work.

any suggestions?

I presume that A,B are sets of nonnegative real numbers, and A/B = { s/
t : s in A, t in B }, with the convention that s/0 = +oo.

Even if s = 0? I don't think so.

Perhaps I should elaborate:

Suppose A = {0} and B = {1/n : n in Z+}. Then A/B = {0} and so
sup(A/B) = 0, while sup(A)/inf(B) = 0/0. Thus, if we wish to have
sup(A/B) = sup(A)/inf(B), we cannot have s/0 = +oo when s = 0. Rather,
we would need 0/0 = 0.

On a different matter: If A or B might be empty, you need to consider how
sup({}) and inf({}) are to be defined.

David W. Cantrell
.

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