Mathematical Analysis I, 2004a

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38 Chapter 2. Real Numbers. Fields upper bound v and a greatest lower bound u. These are abbreviated lub A and glb A and are also called the supremum and infimum of A, respectively; briefly, v =supA, u =infA. However, this assertion, though valid in E 1 , fails to materialize in many other fields such as the field R of all rationals (cf. §§11–12). Even for E 1 ,it cannot be proved from Axioms I through IX. On the other hand, this property is of utmost importance for mathematical analysis; so we introduce it as an axiom (for E 1 ), called the completeness axiom. It is convenient first to give a general definition. Definition 1. An ordered field F is said to be complete iff every nonvoid right-bounded subset A ⊂ F has a supremum (i.e., a lub) in F . Note that we use the term “complete” only for ordered fields. With this definition, we can give the tenth and final axiom for E 1 . X (completeness axiom). The real field E 1 is complete in the above sense. That is, each right-bounded set A ⊂ E 1 has a supremum (sup A) in E 1 , provided A ≠ ∅. The corresponding assertion for infima can now be proved as a theorem. Theorem 1. In a complete field F (such as E 1 ), every nonvoid left-bounded subset A ⊂ F has an infimum (i.e., aglb). Proof. Let B be the (nonvoid) set of all lower bounds of A (such bounds exist since A is left bounded). Then, clearly, no member of B exceeds any member of A, andsoB is right bounded by an element of A. Hence, by the assumed completeness of F , B has a supremum in F ,callitp. We shall show that p is also the required infimum of A, thus completing the proof. Indeed, we have (i) p is a lower bound of A. For, by definition, p is the least upper bound of B. But, as shown above, each x ∈ A is an upper bound of B. Thus (∀ x ∈ A) p ≤ x. (ii) p is the greatest lower bound of A. Forp =supB is not exceeded by any member of B. But, by definition, B contains all lower bounds of A; sop is not exceeded by any of them, i.e., p =glbA =infA. □
§§8–9. Upper and Lower Bounds. Completeness Axiom 39 Note 4. The lub and glb of A (if they exist) are unique. For infA is, by definition, the maximum of the set B of all lower bounds of A, and hence unique, by Note 2; similarly for the uniqueness of sup A. Note 5. Unlike min A and max A, the glb and lub of A need not belong to A. For example, if A is the interval (a, b) inE 1 (a
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150 Chapter 4. Function Limits and
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252 Chapter 5. Differentiation and
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Index Abel’s convergence test, 24
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Index 343 arcwise-, 211 curves as,
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Index 345 as a normed linear space,
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Index 349 limits in one variable, 1
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Index 351 Ordered pair, 1 inverse,
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Index 353 nondecreasing sequences o
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Index 355 Euclidean spaces, 87 norm
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Mathematical Analysis I, 2004a

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