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.

Extreme Value Theorems

The previous proof has a consequence not stated. In the proof, let e = 1. Then there exists x₁,¼,x_n in [a,b] such that if x is in [a,b] then for some j, |x-x_j| £ d and |f(x)-f(x_j)| £ e = 1. The last inequality is equivalent to

Theorem F.6 [Extreme Value Theorem] Suppose b > a. Suppose f(x) is continuous at each point in the interval [a,b]. Then there is a finite interval [K,M] such that f(x) belongs this interval whenever x belongs to the domain [a,b]. Further, there exist a unique pair of real numbers y_min and y_max such y_min £ f(x) £ y_max with the following properties.

 

1. There is at least one real number v in the interval [a,b] such that f(v) = y_min; and
2. there is at least one real number w in the interval [a,b] such that f(w) = y_max.

Proof of Extreme Value Theorem.

As shown above, there is an finite interval [K,M] which contains all the points in the range of f, that is, the set

W = {y |  there exists a point x in [a,b] with y = f(x)}

of the function f. Since W is contained in a finite interval [K,M], a previous theorem implies there is a smallest interval [A,B] Ì [K,M] which contains W.

Let y_min = A = inf(W). Now y_min is either in the range W, or it is a limit point of the range W. In the first case, there is nothing to do. In the second, the closed range theorem implies there is at least one real number v in the interval [a,b] such that f(v) = y_min.

Similarly, let y_max = B = sup(W). Then y_max is either in the range W, or it is a limit point of the range W. In the first case, there is nothing to do. In the second, the closed range theorem implies there is at least one real number v in the interval [a,b] such that f(v) = y_max.

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