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

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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
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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 ).
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