By Terence Tao
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Extra resources for Analysis I (v. 1)
Observe that the numbers ann used in the proof of Theorem 3 form the diagonal of the infinitely extending square composed of all anm . Therefore, the method used above is called the diagonal process (due to G. Cantor). Problems on Countable and Uncountable Sets 1. Prove that if A is countable but B is not, then B − A is uncountable. [Hint: If B − A were countable, so would be (B − A) ∪ A ⊇ B. ] 2. Let f be a mapping, and A ⊆ Df . Prove that (i) if A is countable, so is f [A]; (ii) if f is one to one and A is uncountable, so is f [A].
Problem 4. See also Chapter 1, §9, [Hint: By Theorem 3 of §10, (∃ r ∈ R) √ √ a 2 < r < b 2, r = 0. ) √ Put x = r/ 2; see Problem 4]. 6. Prove that the rational subfield R of any ordered field is Archimedean. [Hint: If x= k p and y = m q (k, m, p, q ∈ N ), then nx > y for n = mp + 1]. 7. Verify the formulas in (1) for powers with positive rational exponents r, s. 8. 4 [Hints: For (i), if r, s > 0 and a > 1, use Problem 9 in §§8–9 to get ar as = sup Aar sup Aas = sup(Aar Aas ). 4 In Problems 8–13, F is assumed complete.
By Definition 1, E 1 is an ordered field. Clearly, whatever follows from the axioms must hold not only in E 1 but also in any other ordered field. Thus we shall henceforth state our definitions and theorems in a more general way, speaking of ordered fields in general instead of E 1 alone. Definition 2. An element x of an ordered field is said to be positive if x > 0 or negative if x < 0. ” 26 Chapter 2. Real Numbers. Fields Definition 3. For any elements x, y of a field, we define their difference x − y = x + (−y).
Analysis I (v. 1) by Terence Tao
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