2.4. SEQUENCES, SEQUENCES OF SETS 73 2.4 Sequences, Sequences of Sets 2.4.1 Sequences Definition 2.4.1 (sequence) Let S R. 1.A sequence in S is a function f : K S where K = n N : n n0 for some n0 N . ! f 2 ≥ 2 g th 2. For each n K, we let xn = f (n). xn is called the n term of the sequence f. 2 For convenience, we usually denote a sequence by xn 1 rather than f. f gn=n0 Some texts also use (x )1 . The starting point, n is usually 1 in which n n=n0 0 case, we simply write xn to denote a sequence. In theory, the starting point does not have to be 1f. However,g it is understood that whatever the starting point is, the elements xn should be defined for any n n0. For example, if the 2n ≥ general term of a sequence is xn = , then, we must have n0 5. We can n 4 ≥ think of a sequence as a list of numbers. In this case, a sequence will look like: xn = x1, x2, x3, ... f Ag sequencef can beg given different ways. 1 2 3 1. List the elements. For example, , , , ... From the elements listed, 2 3 4 the pattern should be clear. n n 2 2. Give a formula to generate the terms. For example, xn = ( 1) . If n! the starting point is not specified, we use the smallest value of n which will work. 3. A sequence can be given recursively. The starting value of the sequence is given. Then, a formula to generate the nth term from one or more previous terms. For example, we could define a sequence by giving: x1 = 2 1 x = (x + 6) n+1 2 n Another example is the Fibonacci sequence defined by: x1 = 1, x2 = 1 xn = xn 1 + xn 2 for n 3 ≥ Like a function, a sequence can be plotted. However, since the domain is a subset of Z, the plot will consist of dots instead of a continuous curve. Since a sequence is defined as a function. everything we defined for functions (bounds, supremum, infimum, ...) also applies to sequences. We restate those definitions for convenience. 74 CHAPTER 2. IMPORTANT PROPERTIES OF R Definition 2.4.2 (Bounded Sequence) As sequence (xn) is said to be bounded above if its range is bounded above. It is bounded below if its range is bounded below. It is bounded if its range is bounded. If the domain of (xn) is n Z : n k for some integer k then the above definition simply state that f 2 ≥ g the set xn : n k must be bounded above, below or both. f ≥ g Definition 2.4.3 (One-to-one Sequences) A sequence (xn) is said to be one- to-one if whenever n = m then xn = xm. 6 6 Definition 2.4.4 (Monotone Sequences) Let (xn) be a sequence. 1. (xn) is said to be increasing if xn xn+1 for every n in the domain of the ≤ sequence. If we have xn < xn+1, we say the sequence is strictly increasing. 2. (xn) is said to be decreasing if xn xn+1 for every n in the domain of the sequence. ≥ 3. A sequence that is either increasing or decreasing is said to be monotone. If it is either strictly increasing or strictly decreasing, we say it is strictly monotone. 2.4.2 Sequences of Sets and Indexed Families of Sets Sequences of sets are similar to sequences of real numbers with the exception that the terms of the sequence are sets. The notation used is also similar. We also generalize the definition to indexed families of sets where the index set is not a subset of Z but any set. More precisely, we have: Definition 2.4.5 Let X R, X = ? and let K = n N : n n0 for some n0 N . A sequence of subsets of X is a function6 f : K fP 2(X), the≥ power set of X. 2 g ! For each n K, we let An = f (n) be a subset of X that is an element of P (x). 2 Example 2.4.6 Consider Nn 1 where Nn = 1, 2, 3, ..., n . Nn 1 is a f gn=1 f g f gn=1 sequence of subsets of N. 1 1 Example 2.4.7 For each n N, define In = x R : 0 < x < = 0, . 2 2 n n In 1 is a sequence of subsets of R. f gn=1 The index we use for the subsets does not have to be an integer. It can be an element of any set. In this case, instead of calling it a sequence of sets, we call it an indexed family of sets. We give a precise definition. Definition 2.4.8 Let A and X be non-empty sets. An indexed family of subsets of X with index set A is a function f : A P (X) (the power set of X). ! Like for sequences, if f : A P (X), then for each A, we let E = f ( ). We use a notation similar! to sequences that is we denote2 the indexed family E A. f g 2 2.4. SEQUENCES, SEQUENCES OF SETS 75 Remark 2.4.9 A sequence of sets is just an indexed family of sets with N as index set. Example 2.4.10 For each x (0, 1), define Ex = r Q : 0 r < x . Then, 2 f 2 ≤ g Ex x (0,1) is an indexed family of subsets of Q. The index set is (0, 1). f g 2 The remainder of this section deals with sequences of sets, though the results and definitions given can be extended to indexed families of subsets. Definition 2.4.11 (Union and Intersection of a Sequence of Subsets) Let An be a sequence of subsets of a set X. f g 1. We define n Ai = A1 A2 ... An [ [ [ i=1 [ = x X : x Ai for some natural number 1 i n f 2 2 ≤ ≤ g Similarly, we define the union of the entire sequence by 1 Ai = x X : x Ai for some natural number i f 2 2 g i=1 [ 2. Similarly, we define n Ai = A1 A2 ... An \ \ \ i=1 \ = x X : x Ai for every natural number 1 i n f 2 2 ≤ ≤ g and 1 Ai = x X : x Ai for every natural number i f 2 2 g i=1 \ 1 Example 2.4.12 Consider Nn 1 where Nn = 1, 2, 3, ..., n . Then, Ni = f gn=1 f g i=1 [ 1 N and Ni = 1 . f g i=1 \ 1 Example 2.4.13 For each n N, define In = x R : 0 < x < . First, 2 2 n we prove that 1 Ii = ? i=1 \ 76 CHAPTER 2. IMPORTANT PROPERTIES OF R 1 If this were not the case, that is if we had x Ii this would mean that no 2 i=1 1 1 \ matter what n is , x < or > x which contradicts the Archimedean principle n n (theorem 2.2.5). We next show that 1 Ii = I1 i=1 [ This because In I1 for any n 1. ≥ Results about finite intersection and union of sets remain true in this set- ting. In other words, we have the equivalent of theorems 1.1.16 (distributive properties), 1.1.20 (De Morgan’slaws), 1.2.33 (direct image of a set) and 1.2.35 (inverse image of a set). We list the theorem here but leave their proof as exercises. In the next few theorems, think of X as R since this is real analysis. Theorem 2.4.14 (Distributive Laws) Let En and E be subsets of a set X. Then, 1 1 1. E Ei = (E Ei) \ \ i=1 ! i=1 [ [ 1 1 2. E Ei = (E Ei) [ [ i=1 ! i=1 \ \ Proof. See problems. Theorem 2.4.15 (De Morgan’sLaws) Let En be a sequence of subsets of X. Then, f g c 1 1 c 1. Ei = Ei i=1 ! i=1 [ \ c 1 1 c 2. Ei = Ei i=1 ! i=1 \ [ Proof. See problems. Theorem 2.4.16 Let f : X Y ! 1. If En is a sequence of subsets of X, then f g 1 1 f Ei = f (Ei) i=1 ! i=1 [ [ 1 1 f Ei f (Ei) i=1 ! i=1 \ \ 2.4. SEQUENCES, SEQUENCES OF SETS 77 2. If Gn is a sequence of subsets of Y , then f g 1 1 1 1 f Gi = f (Gi) i=1 ! i=1 [ [ 1 1 1 1 f Gi = f (Gi) i=1 ! i=1 \ \ Proof. See problems. Definition 2.4.17 (Contracting and Expanding Sequences of Sets) Suppose that An is a sequence of sets. f g 1. We say that An is expanding if we have An An+1 for every n in the domain of thef sequence.g 2. We say that An is contracting if we have An+1 An for every n in the domain off theg sequence. 1 1 Example 2.4.18 Consider the sequence of sets Jn where Jn = , . f g n n 1 1 Jn is a contracting sequence as Jn+1 = , Jn. n+1 n+1 Example 2.4.19 Consider the sequence of sets Kn where Kn = ( n, n). Kn f g is an expanding sequence as Kn Kn+1 = ( (n + 1) , n + 1).