These  Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson

A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

Proofs of  one-sided theorems could be of interest in the study of 2D topology.

Appendix D
Sets and Sequences

Greatest and Least Bounds

A real number M is said to be an upper bound for a set S of real numbers if M ³ s whenever s belongs to the set S. Similarly, a real number K is said to be a lower bound for a set S of real numbers if K £ s whenever s belongs to the set S. The following theorem asserts the existence of a least upper bound B = sup(S) and also a greatest lower bound A = inf(S) for each set S contained in a finite interval [a,b].

Theorem D.1 [On Greatest and Least Bounds] If a set S is contained in a finite interval [a,b] then there is a subinterval [A,B] containing S such that among all intervals of the form [p,q] containing S, the interval [A,B] is the smallest. The number A in this theorem is called the greatest lower bound (The name most positive lower bound might be more appropriate) of the set S. It can be obtained to k decimal places by modifying the proof of the Bolzano-Weierstrass theorem, so that the I_k is the leftmost interval of length 10^-k with the property that there are no points of S to the left of it. The left-end point of I_k then yields A to k decimal places. The decimal expansion defines a number A = inf(S) (read the infimum of S) with the property that no elements of S is to the left of it. In other words, the number A = inf(S) is the greatest number with the property that no numbers in S are lower than it. It is called the greatest lower bound. That is, the number A = inf(S) is lower or £ all numbers in the set S, and it is the largest number with this property. Moreover, by construction, in every interval of the form [A, A+10^-k] where k is a whole number, there must be at least one element of S. Thus, if A is not an element of S, then the set S must be infinite and A is the leftmost limit point of S. On the other hand, if A belongs to S, it is the leftmost or least element of S and S could be finite or infinite. (Exercise: What can be said about the membership of A in the set S when the set S is finite?)

Similarly, the supremum of S is the number sup(S) = B. It is also called the least upper bound (The name least positive upper bound might be more appropriate) of the set S. It can be obtained to k decimal places by modifying the second (or first proof) so that I_k is the rightmost interval of length 10^-k with the property that there are no points of S to the right of it. The left-end point of I_k then yields B to k decimal places. The number B = sup(S) is the leftmost point or number with the property that no element of S is to the right of it. In other words, the number B is the least number with the property that no numbers in S are above it. Here B is above or ³ all numbers in the set S. Moreover, by construction, in every interval of the form [B-10^-k,B] where k is a whole number, there must be at least one element of S. Thus if B is not an element of S, then the set S must be infinite and B is the leftmost limit point of S. On the other hand, if B belongs to S, it is the rightmost or greatest element of S. Here S may be finite or infinite.

Question.   What can be said about the existence of the least and greatest lower bounds of a set S when the latter is contained in a semi-infinite interval of the form [a,¥) or (-¥,b]?

Two Exercises  

1. What can be said about the membership of B in the set S if the set S is finite?
2. Show if S = U ÈV Ì [a,b] then

   sup (S) = max ( sup (U), sup (V))

   and

   inf (S) = min ( inf (U), inf (V))

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