By Terence Tao
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Additional info for Analysis I (Volume 1)
12. 4. 13. 5. 14. 6. Let n be a natural number, and let P(m) be a property pertaining to the natural numbers such that whenever P( m++) is true, then P(m) ·is true. Suppose that P(n) is also true. Prove that P(m) is true for all natural numbers m ::::; n; this is known as the principle of backwards induction. 3 Multiplication In the previous section we have proven all the basic facts that we know to be true about addition and order. To save space and to avoid belaboring the obvious, we will now allow ourselves to use all the rules of algebra concerning addition and order that we are familiar with, without further comment.
10. 1. 2. 2. 3. 3. 5. 4. Prove the identity (a+ b) 2 numbers a, b. 5. 9. ) Chapter 3 Set theory Modern analysis, like most of modern mathematics, is concerned with numbers, sets, and geometry. We have already introduced one type of number system, the natural numbers. We will introduce the other number systems shortly, but for now we pause to introduce the concepts and notation of set theory, as they will be used increasingly heavily in later chapters. ) While set theory is not the main focus of this text, almost every other branch of mathematics relies on set theory as part of its foundation, so it is important to get at least some grounding in set theory before doing other advanced areas of mathematics.
For a:ny natural numbers n and m, n + (m++) = (n+m)++. Again, we cannot deduce this yet from (n++)+m because we do not know yet that a + b = b + a. = (n+m)++ Proof. We induct on n (keeping m fixed). We first consider the base case n = 0. In this case we have to prove 0 + (m++) = (0 + 4 From a logical point of view, there is no difference between a lemma, proposition, theorem, or corollary - they are all claims waiting to be proved. However, we use these terms to suggest different levels of importance and difficulty.