{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

seqnotes

# seqnotes - NOTES ON SUPS INFS AND SEQUENCES LANCE D DRAGER...

This preview shows pages 1–3. Sign up to view the full content.

NOTES ON SUP’S, INF’S AND SEQUENCES LANCE D. DRAGER The purpose of these notes is to briefly review some material on sup’s, inf’s and sequences from undergraduate real analysis, to give an introduction to using these concepts in the extended real numbers, and to give an exposition of lim sup and lim inf. 1. Sup’s and Inf’s in the Real Numbers I’ll use the symbol R to denote the set of real numbers. Let A be a nonempty subset of R . A number u is an upper bound for A if a u for all a A . We say that A is bounded above if it has an upper bound. Note that saying that a number t is not an upper bound for A is equivalent to the statement “There is some a A such that t < a .” A number s is called the supremum of A (sup) if it is the least upper bound of A , i.e., s is an upper bound for A and if u is an upper bound for A then s u . We denote the supremum of A by sup( A ). (A little thought shows there can be at most one number that satisfies the definition of sup.) As you recall, the real numbers are constructed by “filling in the holes” in the rational numbers. One way of saying that the holes have all been filled is the Completeness Axiom for the Real Numbers, which is stated as follows. Completeness Axiom for the Real Numbers. If A is a nonempty subset of R that is bounded above then A has a supremum. We have similar concepts when working on the other side of A . Again, let A be a nonempty subset of R . A number is a lower bound for A if a for all a A . We say A is bounded below if it has a lower bound. A number i is the infimum of A if it is the greatest lower bound of A , i.e., i is a lower bound and if is a lower bound for A , then i . It is not necessary to add an axiom about inf’s to the definition of the real numbers, since the existence of inf’s can be deduced from the existence of sup’s by what I like to call “The Reflection Trick.” Here we set - A = {- a | a A } . Proposition 1.1. [Reflection Trick] Let A be a nonempty subset of R . (1) If A is bounded below, - A is bounded above and inf( A ) = - sup( - A ) . (2) If A is bounded above, - A is bounded below and sup( A ) = - inf( - A ) . Exercise 1.2. Prove the last Proposition. The following Propositions are frequently used in proofs involving sup and inf. Proposition 1.3. Let A be a nonempty subset of R . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 LANCE D. DRAGER (1) Suppose that A is bounded above. Then α < sup( A ) if and only if α < a for some a A . (2) Suppose that A is bounded below. Then inf( A ) < β if and only if a < β for some a A . Proof. To prove the first statement, assume first that α < sup( A ). Since α is less than the least upper bound of A , α is not an upper bound for A . Thus, there is some a A so that α < a . For the second part of the proof, assume that α < a for some a A . Then α < a sup( A ) (since sup( A ) is an upper bound), so α < sup( A ).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern