Introduction to Analysis Constructing R from Q

Introduction to Analysis Constructing R from Q Definition. A subset A ⊂ Q is called a cut if it posses the following three properties 1. A 6= ∅ and A 6= Q. 2. If r ∈ A, then also A contains every rational q < r. 3. A does not have a maximum, that is, if r ∈ A, then there exists an s ∈ A with r < s. Remark. Note that for a cut A, if r ∈ A and s 6∈ A, then r < s. This is because all rational q < r are in A. Hence, a rational number s 6∈ A must satisfy r < s. Example. 1. Cr = {t ∈ Q | t < r} √ √ 2. {t ∈ Q | t < 2} = {t ∈ Q | t ≤ 2 3. Not a cut {t ∈ Q | t ≤ 2}. Definition. The real numbers, denoted R, is the set of all cuts in Q. 1 Ordering on Cuts Theorem. (Ordering on R) Inclusion of sets (⊆) is an ordering on cuts. Proof. We need to prove the three properties of an ordering: 1. For arbitrary cuts A, B ∈ R at least one of A ⊆ B or B ⊆ A is true. Proof. If A ⊆ B then we are done, so assume not. We need to show that B ⊆ A. By assumption, there exists an element a ∈ A such that a 6∈ B. Let b ∈ B be arbitrary. Since a 6∈ B the lemma implies that b < a and by definition of a cut we must have b ∈ A. 2. If A ⊆ B and B ⊆ A then A = B. Proof. This is standard definition of equality of sets. 3. (transitive) If A ⊆ B and B ⊆ C then A ⊆ C. Proof. Again, trivial since we are working with sets. 1 2 Addition on Cuts Theorem. (Addition in R) Given A, B, C ∈ R we define 1. A + B = {a + b | a ∈ A, b ∈ B} such that A + B = B + A and (A + B) + C = A + (B + C). 2. O = {t ∈ Q | t < 0} such that A + O = A. 3. −A = {r ∈ Q | there exists t 6∈ A with t < −r} such that A + (−A) = O 4. If A ⊆ C then A + B ⊆ A + C Proof. Let A, B be two cuts. 1. (Addition) We first need to see that A + B is a cut, then we need to check that addition satisfies: commutativity, associativity, identities exist, and inverses exist. (a) (A + B is a cut): i. First we see that A + B is non-empty since A, B are non-empty. To see that A + B 6= Q we find an upper bound for A + B. Let l1 6∈ A and l2 6∈ B. For any a ∈ A and b ∈ B we have that a < l1 and b < l2 . Hence we have l1 + l2 as an upper bound of A + B. ii. Let r = a + b ∈ A + B. Let s ∈ Q such that s < r. Then we have s < a + b and hence s − b < a. Hence s − b ∈ A. So we have s = (s − b) + b ∈ A + B. iii. To see that A + B does not have a maximum, fix c ∈ A + B and write c = a + b for some a ∈ A and b ∈ B. So there exists and s ∈ A with a < s and an r ∈ B with b < r. So we have c = a + b < s + r. (b) (Commutative): We simply write A + B = {a + b | a ∈ A, b ∈ B} = {b + a | a ∈ A, b ∈ B} = B + A. (c) (Associative): Let C be a third cut. Then we simply write (A + B) + C = {(a + b) + c | a ∈ A, b ∈ B, c ∈ C} = {a + (b + c) | a ∈ A, b ∈ B, c ∈ C} = A + (B + C). 2. (Identity): Let O = {p ∈ Q | p < 0}. (a) O is clearly a cut. (b) We need to show that A + O ⊆ A and A ⊆ A + O. Let a + b ∈ A + O. Then we have a + b < a ∈ A. Similarly, let a ∈ A. Then for some ² > 0, s + ² ∈ A. Then (s + ²) − (−²) ∈ A + B. 3. (Additive Inverse): Let −A = {r ∈ Q | ∃t 6∈ A with t < −r}. (a) (−A is a cut): Note that for A = {r ∈ Q | r < 1/2 we have that −A = {r ∈ Q | r < −1/2}. 2 i. Since A 6= Q there exists a t 6∈ A. Since t < t + 1 we can conclude that −(t + 1) ∈ −A by the definition of −A, so we have −A is non-empty. ii. To show −A 6= Q we know that A is non-empty so there exists an a ∈ A. If r ∈ −A, there exists a t 6∈ A with t < −r. Then t 6∈ A implies that a < t and hence r < −a. So −A is bounded above by −a. iii. Let r ∈ −A and consider s ∈ Q with s < r. We need s ∈ −A. Because r ∈ −A there exists a t 6∈ A with t < −r. Since s < r implies −r < −s we have t < −s and hence s ∈ −A. iv. Let r ∈ −A, then there exists a t 6∈ A with t < −r. By the density of Q we can choose an s ∈ Q such that t < s < −r. Hence −s ∈ −A and because r < −s, we see that −A does not possess a maximum. (b) (A + −A = O): Let r ∈ −A. Because −r ∈ −A we know there exists a t 6∈ A with t < −r. Since t 6∈ A we have a < t for all a ∈ A, and hence a < −r. Thus, a + r < 0 and so a + r ∈ O as needed to show that A + (−A) ⊆ O. To show the reverse inclusion let o ∈ O and we need to find an a ∈ A and a b ∈ −A such that −o a + b = o. Let ² = |o| 2 = 2 . Now choose t ∈ Q such that t 6∈ A and t − ² ∈ A. t 6∈ A implies that −(t + ²) ∈ −A. So we have −(t + ²) + (t − ²) = −2² = o. 4. (Respects ordering): Assume that A ⊆ B. Then consider r = a + c ∈ A + C. Then we have r ∈ B + C since a ∈ B by assumption. 3 Multiplication in R Theorem. (Multiplication in R) Let A, B ≥ O be in R. 1. For A, B ⊇ O, AB = {ab | a ∈ A, b ∈ B with a, b ≥ 0} ∪ {q ∈ Q, q < 0} 2. I = {t ∈ Q | t < 1} such that AI = A. 3. For A 6= O, A−1 = {t ∈ Q | 1 t 6∈ A, ∃r 6∈ As.t.r < 1t } ∪ {t ∈ Q | t ≤ 0} such that AA−1 + I. 4. C(A + B) = CA + CB. 5. AC ⊇ O. Remark. For negative multiplication and inverse, we take −A. Proof. 1. (AB is a cut) (a) AB is not empty because all rational numbers t < 0 are in AB. Let s, t be upper bounds for A and B respectively. Then AB is bounded above by st, so AB cannot be all of Q. 3 (b) Let t ∈ AB be arbitrary and let s ∈ Q satisfy s < t. If s < 0 the s ∈ AB by the definition of our product, so let 0 ≤ s < t. So we have t = ab with a ∈ A and b ∈ B and a, b > 0. Because s < ab we have sb < a and hence sb ∈ A. Then ³s´ s= (b) ∈ AB. b (c) Let t ∈ AB. If t < 0 then t < t/2 and t/2 ∈ AB since t/2 < 0 as well. If t ≥ 0, then t = ab for some a ∈ A and b ∈ B. Applying the 3rd property of cuts to A, B yields s ∈ A and r ∈ B such that a < s and b < r. So we have sr ∈ AB and ab < sr. 2. (Multiplicative Identity): (a) We first show that I is a cut i. This is easy since 0 ∈ I and 2 6∈ I. ii. Assume that t ∈ I, then for s < t we have s < 1 and hence s ∈ I. iii. Let t ∈ I. If t < 0, then t < t/2 ∈ I. if t ≥ 0 then t < t+1 2 ∈ I. (b) (AI ⊆ A): Let q ∈ AI. Because I ⊇ O, then either q < 0 or q = ab with a, b ≥ 0 and a ∈ A and b < 1. If q < 0, then q ∈ A since A ≥ O. In the other case, q = ab < a and hence ab ∈ A. (c) (A ⊆ AI): Let a ∈ A. If a < 0 then a ∈ AI by the product of cuts. If a ≥ 0, then we can pick a t ∈ A with a < t. Hence at < 1 and hence at ∈ I. So we have ³a´ a = (t) ∈ AI. t 3. (Multiplicative Inverse) (a) A−1 is a cut. i. First, −1 ∈ A−1 so it is not empty. So we need to show it is not all of Q. Let 0 ≤ t 6= 0 ∈ A (this is possible, since A 6= O and A ⊇ O). Then 1t 6∈ A−1 . ii. Let t ∈ A−1 . Assume that s < t. Then we have 1t < 1s and that 1t 6∈ A so we must have 1 1 −1 . s 6∈ A and hence s ∈ A iii. Let t ∈ A−1 . If t < 0, let s = 2t . So assume t ≥ 0. Then 1t 6∈ A and there exists and r 6∈ A such that r < 1t . Since Q is dense, there exists a rational number r < s < 1t that is also not in A (If t = 0 choose any s > r.). Then we have that 1s 6∈ A and that r < s, so we have 1s ∈ A−1 and t < 1s . (b) AA−1 ⊆ I Let t ∈ AA−1 with t ≤ 0, then t ∈ I. Let a ∈ A and b ∈ A−1 with a, b > 0. (Note that a = b = 1 is a contradiction, since that would imply that 1b ∈ A, so we must have one of less than 1.) Since 1b 6∈ A we have a < 1b and hence µ ¶ 1 ab < b=1 b Hence ab ∈ I. 4 (c) I ⊆ AA−1 If A = I then A−1 = I so there is nothing to show, so assume not. WLOG assume that A ⊆ I ⊆ A−1 . Let r ∈ I. If r ≤ 0, then r ∈ AA−1 . Now assume that we 1 have 0 < r < 1. There exits an n ∈ N such that rn ∈ A and rn−1 6∈ A. If rn−1 ∈ A−1 we have µ ¶ 1 n r=r rn−1 1 and are done. So assume that rn−1 6∈ A−1 . This means that for every ² > 0 we have n−1 r − ² ∈ A. So we can find a rational ² > 0 such that rn < r(rn−1 + ²) < rn−1 1 −1 by the density of the rational numbers. So we have rn−1 since (rn−1 + ²) 6∈ A and +² ∈ A rn−1 < rn−1 + ². But also note that r(rn−1 + ²) ∈ A. r = r(rn−1 + ²) 1 rn−1 +² ∈ AA−1 4. (Distributive Property): We show inclusion in both ways. (a) If t ∈ C(A + B) then we can write t = c(a + b). Since these are all rational numbers we have t = ca + cb ∈ CA + CB. (b) If t ∈ CA + CB then we can write t = ca + cb. Since these are all rational number we have t = c(a + b) ∈ C(A + B). 5. (Respects ordering): By the definition of AB this is trivial. Remark. You will show for homework the embedding Q ⊆ R: Let Cr = {t ∈ Q | t < r}. Then we have 1. Cr + Cs = Cr+s 2. Cr Cs = Crs . 3. etc. For r ∈ Q this provides Q and a subfield of R (as cuts). 4 Least Upper Bounds Definition. A set A ⊆ R is bounded above if there exists a B ∈ R such that A ≤ B for all A ∈ A. B is called an upper bound for A. S ∈ R is a least upper bound for A if it satisfies 1. S is an upper bound for A. 2. If B is any upper bound for A, then S ≤ B. Theorem. Let S = ∪A∈A A. S is the least upper bound for A. 5 Proof. 1. First we show that S is a cut. (a) Since A 6= ∅, then S 6= ∅. Since A is bounded above by some cut B we have S ≤ B. Since B 6= Q we have S 6= Q. (b) Let a ∈ S and r ∈ Q with r < a. Since a ∈ S, then a ∈ A for some A ∈ S. Hence we have r ∈ A and hence r ∈ S. (c) Let a ∈ S, then a ∈ A for some A ∈ S. Therefore, there is an r ∈ A such that a < r and since r ∈ A, then r ∈ S. 2. Now we need to show that it is in fact a least upper bound. (a) S is an upper bound for A since A ≤ S for all A ∈ A. (b) Let B be an arbitrary upper bound for A. We need S ≤ B. Let s ∈ S. There fore s ∈ A for some A ∈ A and hence s ∈ B. Therefor S ≤ B. Since least upper bounds exists and R is complete. R is also an ordered field containing Q. Remark. Cantor used equivalence classes of Cauchy sequences to define R. (two Cauchy sequences are equivalent if (xn − yn ) → 0. Remark. Cuts may seem a weird concept compared to a decimal number, but with decimals how do you resolve the fact that .5 = .499. 6