6 Subsequences and sequential compactness 6.1 Nested intervals and - - PDF document

6 Subsequences and sequential compactness

6.1 Nested intervals and nested d-cells

Recall the nested intervals principle from G11ACF: Let Ik = [ak, bk] ⊆ R be closed (non-empty) intervals in R with I1 ⊇ I2 ⊇ I3 ⊇ . . . (such a sequence of sets is said to be nested decreasing). Then there exists at least one x ∈ R such that x is in all

• f the intervals Ik, i.e., such that x ∈ ∞

k=1 Ik.

(This fails for non-closed intervals in general. See question sheets.) Moreover, if bk − ak → 0 as k → ∞, then this point x is unique. Gap to fill in

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Before discussing the d-dimensional analogue of this result, let us first look at the condition that bk − ak → 0 as k → ∞. Note that bk − ak = max{|x − y| : x, y ∈ [ak, bk] } . In general, for a non-empty, bounded subset A of Rd, we define the diameter of A, diam (A), by diam (A) = sup{x − y : x, y ∈ A } the supremum of the possible distances between pairs of points of A. From earlier results we know that 0 ≤ diam (A) < ∞ for non-empty bounded sets. For (non-empty, bounded) closed d-cells, this supremum is actually a maximum, and this maximum is achieved by considering any pair of diametrically opposite corners of the d-cell. Gap to fill in

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Traditionally, when authors discuss ‘closed intervals’ in R, they mean intervals of the form [a, b] where a and b are real numbers with a ≤ b, and not unbounded intervals such as [0, ∞[. For the rest of this chapter, whenever we discuss ‘closed intervals’ (or ‘closed d-cells’), we will follow this traditional terminology, so that we will mean non-empty, closed and bounded intervals (or d-cells). We now come to the d-dimensional analogue of the nested intervals principle. Theorem 6.1.1 (Nested d-cells principle) Let d ∈ N, and let Ik be a nested decreasing sequence of closed d-cells (so I1 ⊇ I2 ⊇ I3 ⊇ . . . ). Then there exists at least one point x ∈ Rd such that x is in all of the d-cells Ik, i.e., such that x ∈ ∞

k=1 Ik.

Moreover, if diam (Ik) → 0 as k → ∞ then this point x is unique. The statement and applications of this this d-dimensional principle are examinable, but the proof is an NEB exercise. It is, in fact, fairly easy to deduce the result from the nested intervals principle in R.

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6.2 The Bolzano–Weierstrass Theorem

Let x1, x2, x3, x4, x5, . . . be a sequence in Rd. We can form a new sequence by omitting some of the sequence members, e.g. (i) x1, x3, x5, x7, . . . (ii) x2, x4, x6, x8, . . . (iii) x2, x32, x532, x7532, . . . (All three examples are meant to continue indefinitely.) In (i) we have formed the new sequence (yn), where yn = x2n−1. In (ii) we have formed (zn), where zn = x2n. In (iii) it is perhaps less clear what the pattern is but in any case we have chosen a strictly increasing sequence of positive integers k1 = 2 < k2 = 32 < k3 = 532 < k4 = 7532 < . . . and formed a new sequence (pn) given by pn = xkn.

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Definition 6.2.1 A subsequence of a sequence (xn) in Rd is a sequence of the form (yn) = (xkn), where k1 < k2 < k3 < . . . is an increasing sequence of positive integers. This means: when a subsequence is formed the order of the sequence members is kept and repetitions of the index are excluded. There may be repeated values, but only if there were repeated values in the original sequence. Why is this notion important? It happens often that divergent (i.e. non-convergent) sequences have convergent subsequences. For example, xn = (−1)n defines a divergent sequence in R but (x2n) is convergent. What about the sequence xn = sin(n) in R? (Here we are working in radians, as usual.) The Bolzano–Weierstrass Theorem (below) will show, in particular, that this sequence also has at least one convergent subsequence.

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We say that a sequence (xn) in Rd is bounded if the set {xn | n ∈ N} ⊆ Rd of all sequence members is bounded, i.e. there exists M > 0 such that, for all n ∈ N, we have ||xn|| ≤ M. Now we can formulate the famous Theorem 6.2.2 (Bolzano–Weierstrass Theorem) Every bounded sequence (xn) in Rd has at least one convergent subsequence. [ So, in particular, (sin(n)) ⊆ R has a convergent subsequence!] The statement of the Bolzano–Weierstrass Theorem and its applications are examinable. The proof is NEB. See books for full details, if interested. Most books give a fairly straightforward proof based on the nested d-cells principle, or some similar result.

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6.3 Sequential compactness

The notion of sequential compactness and other related notions of compactness are extremely important in analysis and mathematics in general and have far-reaching applications. Definition 6.3.1 A subset E ⊆ Rd is said to be sequentially compact if every sequence (xn) ⊆ E has at least one subsequence which converges to a point of E. We can characterize sequentially compact subsets of Rd as follows. Theorem 6.3.2 (Heine-Borel Theorem, sequential compactness version) Let E ⊆ Rd be any subset. Then the following conditions are equivalent: (i) E is sequentially compact; (ii) E is closed and bounded. Gap to fill in

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Example 6.3.3

• 1. With the traditional terminology,

closed intervals in R and closed d-cells and closed balls in Rd are sequentially compact.

• 2. The set {0} ∪ { 1

n | n ∈ N} is sequentially compact.

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Sequentially compact sets are ‘nice’ sets in many respects, as we will see. It also turns out that, for subsets of Rd, sequential compactness is equivalent to a topological condition involving coverings by open sets. This related condition, which is called simply compactness, is of great importance throughout mathematics. See books and/or the module G13MTS if interested.

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