Supremum and Infimum graphs

[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.

 

[HallsofIvy]

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

 

[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.

 

[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, a_n=n^{(-1)^n}

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

 

[Bacle2]

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, a_n=n^{(-1)^n}

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

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.