Supremum and Infimum graphs

1. Mar 8, 2012

elizaburlap

In class, we have been introduced to the supremum and infimum concepts and shown them on graphs, but I am wondering how to go about deriving them, and determining if they are part of the set, without actually having to graph them- especially for more complicated sets.

2. Mar 8, 2012

HallsofIvy

Staff Emeritus
Are you able to find "upper bound" and "lower bounds" for sets?

3. Mar 8, 2012

Bacle2

Some ideas:

Have you already seen limit points? If so, try showing that both the sup and the inf (when both are finite*) , are limit points of a set. Then look for a charcterization of closed sets in terms of limit points.

*This can be extended to the infinite case too, but let's start slowly.

4. Mar 9, 2012

elizaburlap

How would you go about extending it to infinity?

In the text it has a few examples that span n from 1 to infinity.

Such as, an=n(-1)^n

I understand that it does converge, because an approaches 0 as n approaches infinity, but when the equations become more complicated, how to I recognise this without a graph?

5. Mar 9, 2012

Bacle2

I actually used 'limit point' here a little too losely (specially since ∞ is not a real number); what I meant is that , in the case the sup is ∞ , the values would become indefinitely-large. In the extended reals, every 'hood (neighborhood) of ∞ would contain points of the set.

To recognize/determine the limit, I would suggest looking at the expression and trying to understand what happens with it as you approach ∞. Does it oscillate, increase, etc. If you cannot tell right away, consider trial-and-error. Assume a certain value is the Sup (Inf) , and put it to the test. That is the best I got; I cannot think of any sort of algorithm. It just seems to come down to practicing.