8. SEQUENCES AND SUBSEQUENCES 11 And why this theorem is important for us? Because many economic problems are concerned with finding a maximal (or a minimal) value of a function on some set. Weierstrass theorem provides conditions under which such search is meaningful!!! This theorem and its implications will be much dwelt upon later in the notes, so we just give here one example. The consumer utility maximisation problem is the problem of finding the maximum of utility function subject to the budget constraint. According to Weierstrass theorem, this problem has a solution if utility function is continuous and the budget set is compact. 8. Sequences and Subsequences Let us consider again some metric space (X, d). An infinite sequence of points in (X, d) is simply a list x1 , x2 , x3 , . . . , where . . . indicates that the list continues “forever.” We can be a bit more formal about this. We first consider the set of natural numbers (or counting numbers) 1, 2, 3, . . . , which we denote N. We can now define an infinite sequence in the following way. Definition 6. An infinite sequence of elements of X is a function from N to X. Notation. If we look at the previous definition we see that we might have a sequence s : N → X which would define s(1), s(2), s(3), . . . or in other words would define s(n) for any natural number n. Typically when we are referring to sequences we use subscripts (or sometimes superscripts) instead of parentheses and write s1 , s2 , s3 , . . . and sn instead of s(1), s(2), s(3), . . . and s(n). Also rather than saying that s : N → X is a sequence we say that {sn } is a sequence or even that {sn }∞ n=1 is a sequence. Lets now examine a few examples. Example 4. Suppose that√(X, d) is R the real numbers with the usual metric d(, x, y) = |x − y|. Then {n}, { n}, and {1/n} are sequences. Example 5. Again, suppose that (X, d) is R the real numbers with the usual metric d(x, y) = |x − y|. Consider the sequence {xn } where ( 1 if n is odd xn = 0 if n is even √ We see that {n} and { n} get arbitrary large as n gets larger, while in the last example xn “bounces” back and forth between 0 and 1 as n gets larger. However for {1/n} the element of the sequence gets closer and closer to 0 (and indeed arbitrarily close to 0). We say, in this case, that the sequence converges to zero or that the sequence has limit 0. This is a particularly important concept and so we shall give a formal definition. Definition 7. Let {xn } be a sequence of points in (X, d). We say that the sequence converges to x0 ∈ X if for any ε > 0 there is N ∈ N such that if n > N then d(xn , x0 ) < ε. Informally we can describe this by saying that if n is large then the distance from xn to x0 is small. If the sequence {xn } converges to x0 , then we often write xn → x0 as n → ∞ or limn→∞ xn = x0 . 12 1. LOGIC, SETS, FUNCTIONS, AND SPACES Exercise 34. Show that if the sequence {xn } converges to x0 then it does not converge to any other value unequal to x0 . Another way of saying this is that if the sequence converges then it’s limit is unique. We have now seen a number of examples of sequences. In some the sequence “runs off to infinity;” in others it “bounces around;” while in others it converges to a limit. Could a sequence do anything else? Could a sequence, for example, settle down each element getting closer and closer to all future elements in the sequence but not converging to any particular limit? In fact, depending on what the space X is this is indeed possible. First let us recall the notion of a rational number. A rational number is a number that can be expressed as the ratio of two integers, that is r is rational if r = a/b with a and b integers and b 6= 0. We usually denote the set of all rational numbers Q (since we have already used R for the real numbers). We now consider and example in which the underlying space X is Q. Consider the sequence of rational numbers defined in the following way x1 = 1 xn + 2 . xn+1 = xn + 1 This kind of definition is called a recursive definition. Rather than writing, as a function of n, what xn is we write what x1 is and then what xn+1 is as a function of what xn is. We can obviously find any element of the sequence that we need, as long as we sequentially calculate each previous element. In our case we’d have x1 = 1 1+2 3 = = 1.5 x2 = 1+1 2 3 + 2 7 x3 = 23 = = 1.4 5 2 +1 x4 = x5 = x6 = 7 17 5 +2 = ≈ 1.416667 7 12 + 1 5 17 41 12 + 2 = ≈ 1.413793 17 29 12 + 1 41 99 29 + 2 ≈ 1.414286 = 41 70 29 + 1 .. . We see that the sequence goes up and down but that it seems to be “converging.” What is it converging to? Lets suppose that it’s converging to some value x0 . Recall that xn + 2 . xn+1 = xn + 1 We’ll see later that if f is a continuous function then lim n → ∞f (xn ) = f (lim n → ∞xn ). In this case that means that xn + 2 x0 = lim n → ∞xn+1 = lim n → ∞ xn + 1 x0 + 2 = . x0 + 1 Thus we have x0 + 2 x0 = x0 + 1 8. SEQUENCES AND SUBSEQUENCES 13 √ and if we solve this we obtain x0 = ±√ 2. Clearly if xn > 0 then √ xn+1 >√0 so our sequence can’t be converging to − 2 so we must have x0 = 2. But 2 is not in Q. Thus we have a sequence of elements in Q that are getting very close to each other but are not converging to any element of Q. (Of course the sequence is converging to a point in R. In fact one construction of the real number system is in terms of such sequences in Q. Definition 8. Let {xn } be a sequence of points in (X, d). We say that the sequence is a Cauchy sequence if for any ε > 0 there is N ∈ N such that if n, m > N then d(xn , xm ) < ε. Exercise 35. Show that if {xn } converges then {xn } is a Cauchy sequence. A metric space (X, d) in which every Cauchy sequence converges to a limit in X is called a complete metric space. The space of real numbers R is a complete metric space, while the space of rationals Q is not. Exercise 36. Is N the space of natural or counting numbers with metric d given by d(x, y) = |x − y| a complete metric space? In Section 6 we defined the notion of a function being continuous at a point. It is possible to give that definition in terms of sequences. Definition 9. Suppose (X, dX ) and (Y, dY ) are metric spaces, x0 ∈ X, and f : X → Y is a function. Then f is continuous at x0 if for every sequence {xn } that converges to x0 in (X, dX ) the sequence {f (xn )} converges to f (x0 ) in (Y, dY ). Exercise 37. Show that the function f (x) = (x + 2)/(x + 1) is continuous at any point x 6= −1. Show that this means that if xn → x0 as n → ∞ then x0 + 2 xn + 2 = . lim n→∞ xn + 1 x0 + 1 We can also define the concept of a closed set (and hence the concepts of open sets and compact sets) in terms of sequences. Definition 10. Let (X, d) be a metric space. A set S ⊂ X is closed if for any convergent sequence {xn } such that xn ∈ S for all n then limn→∞ xn ∈ S. A set is open if its complement is closed. Given a sequence {xn } we can define a new sequence by taking only some of the elements of the original sequence. In the example we considered earlier in which xn was 1 if n was odd and 0 if n was even we could take only the odd n and thus obtain a sequence that did converge. The new sequence is called a subsequence of the old sequence. Definition 11. Let {xn } be some sequence in (X, d). Let {nj }∞ j=1 be a sequence of natural numbers such that for each j we have nj < nj+1 , that is n1 < n2 < n3 < . . . . The sequence {xnj }∞ j=1 is called a subsequence of the original sequence. The notion of a subsequence is often useful. We often use it in the way that we briefly referred to above. We initially have a sequence that may not converge, but we are able to take a subsequence that does converge. Such a subsequence is called a convergent subsequence. Definition 12. A subset of a metric space with the property that every sequence in the subset has a convergent subsequence is called sequentially compact. Theorem 3. In any metric space any compact set is sequentially compact. 14 1. LOGIC, SETS, FUNCTIONS, AND SPACES If we restrict attention to finite dimensional Euclidian spaces the situation is even better behaved. Theorem 4. Any subset of Rn is sequentially compact if and only if it is compact. Exercise 38. Verify the following limits. n (i) lim =1 n→∞ n + 1 n+3 =0 (ii) lim 2 n→∞ √ n +1 √ (iii) lim n + 1 − n = 0 n→∞ √ n (iv) lim an + bn = max{a, b} n→∞ Exercise 39. Consider a sequence {xn } in R. What can you say about the sequence if it converges and for each n xn is an integer. Exercise 40. Consider the sequence 1 1 2 1 2 3 1 2 3 4 1 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, . . . . For which values z ∈ R is there a subsequence converging to z? Exercise 41. Prove that if a subsequence of a Cauchy sequence converges to a limit z then so does the original Cauchy sequence. Exercise 42. Prove that any subsequence of a convergent sequence converges. Finally one somewhat less trivial exercise. Exercise 43. Prove that if limn→∞ xn = z then x1 + · · · + xn =z lim n→∞ n 9. Linear Spaces The notion of linear space is the axiomatic way of looking at the familiar linear operations: addition and multiplication. A trivial example of a linear space is the set of real numbers, R. What is the operation of addition? The one way of answering the question is saying that the operation of addition is just the list of its properties. So, we will define the addition of elements from some set X as the operation that satisfies the following four axioms. A1: x + y = y + x for all x and y in X. A2: x + (y + z) = (x + y) + z, for all x, y, and z in X. A3: There exists an element, denoted by 0, such that x + 0 = x for all x in X. A4: For every x in X there exist an element y in X, called inverse of x, such that x + y = 0. And, to make things more interesting we will also introduce the operation of ‘multiplication by number’ by adding two more axioms. A5: 1x = x for all x in X. A6: α(βx) = (αβ)x for all x in X and for all α and β in R. Finally, two more axioms relating addition and multiplication. A7: α(x + y) = αx + αy for all x and y in X and for all α in R. A8: (α + β)x = αx + βx for all x in X and for all α and β in R.