Real analysis is a beacon of stability in the otherwise unpredictable evolution of the math curriculum. Unlike the main body of the book, much effort has gone into reviewing the exercises and projects.
Some Preliminaries
Because of the nested property of this particular collection of sets, it's not that hard to see. Induction is used to prove that a certain statement holds for every value of n ∈ N, but this does not imply the validity of the infinite case.
The Axiom of Completeness
Although the notation s = lubA is sometimes used, we will always write s = supA for the smallest upper bound. This is equivalent to a≤b−c for alla∈A, from which we conclude that b−ci is an upper bound for A.
Consequences of Completeness
The setQ is an extension of N and Rin turn is an extension of Q. The next few results show how N and Q are in R. i) Given any number x ∈ R, there exists an n∈N satisfying n > x. The first step is to choose the denominator so large that successive increments of magnitude 1/ are barely too close together to step over the interval (a, b).
Cardinality
We will now use the nested interval property (Theorem 1.4.1) to produce a real number that is not there. The inescapable conclusion is that, despite having encountered so few of them, the irrational numbers form a far larger subset of RthanQ. The properties of countable sets described in this discussion are useful for a few exercises in future chapters.
Cantor’s Theorem
Cantor's theorem 35 The assumption that f is on means that every subset of A appears as f(a) for an a∈A. Cantor's theorem states that there is no on-function of NtoP(N); in other words, the power set of the natural numbers is uncountable.
Epilogue
The proof that no such algorithm exists uses a diagonalization-type construction at the heart of the argument. As a more immediate example of this phenomenon, the diagonalization method is used again in Chapter 6 – constructively – as a crucial step in the proof of the Arzela-Ascoli theorem.
The Limit of a Sequence
Significant insight into the role of the quantifiers in defining convergence can be gained by studying an example of a sequence that does not have a limit. At every college in the United States, there is a student who is at least seven feet tall.
The Algebraic and Order Limit Theorems
Our goal is to find a point in the sequence (can) we have after. Although there are some dangers to avoid (see Exercise 2.3.7), the Algebraic Limit Theorem verifies that the relation between algebraic combinations of sequences and the limiting process is as problem-free as we could hope for.
The Monotone Convergence Theorem and a First Look at
Because the terms in the sum are all positive, the sequence of partial sums given by Thus, 2 is an upper bound for the sequence of partial sums, so by the Monotone Convergence Theorem, . For each series, find an explicit formula for the sequence of partial sums and determine whether the series converges.
Subsequences and the Bolzano–Weierstrass Theorem
Let (an) be a bounded sequence such that there exists M > 0 satisfying. The midpoint is included in both halves.) Now it must be that at least one of these closed intervals contains an infinite number of terms in the sequence (an). The Bolzano–Weierstrass theorem is extremely important, and so is the strategy used in the proof. Show that there exists a subsequence (ank) that converges tos= supS. This is a direct proof of the Bolzano–Weierstrass theorem using the Axiom of Completeness.).
The Cauchy Criterion
The significance of the definition of the Cauchy sequence is that no limit is mentioned. We have already been in this situation in the proofs of the monotonic convergence theorem and the Bolzano–Weierstrass theorem. Finally, we needed BW in our proof of the Cauchy Criterion (CC) for convergent sequences.
Properties of Infinite Series
It is important that all the original terms appear in a new order over time and that no term is repeated. In Section 2.1, we constructed a special rearrangement of the alternating harmonic series that converges to a limit different from that of the original series. Consider this (fictional) definition for a moment and then decide which of the following statements are valid claims about subvergent species:. b) All convergent species are subvergent.
Double Summations and Products of Infinite Series
To prove the theorem, we must show that the two iterated sums converge to the same limit. Note that the same argument can be used once it has been established that for any fixed column j the sum∞. i=1aij converges to a real number cj. If the two series being multiplied absolutely converge, it is not too difficult to prove that the sum can be calculated in the most appropriate way.
Epilogue
There is some strong evidence that there is not much left when the total length of the removed intervals is considered. For a square, increasing each length by a factor of 3 results in a larger square containing 9 copies of the original square. Finally, the augmented cube yields a cube that contains 27 copies of the original cube in its volume.
Open and Closed Sets
This finding leads to the following result. i) The union of any collection of open sets is open. ii) The intersection of a finite collection of open sets is open. This is because any neighborhood centered at zero, no matter how small, will contain points A. The set F =A∪ {0} is an example of a closed set and is called the closure of A. We will discuss the closure of a set in moment.). ii) Prove that a closed interval. E for any set E⊆R. The last theorem of this section should be compared with Theorem 3.2.3. i) The union of a finite collection of closed sets is closed. ii) The intersection of any collection of closed sets is closed.
Compact Sets
All of the following statements are equivalent in the sense that each of them implies the other two:. ii) K is closed and bounded. iii) Every open cover for K has a finite subcover. Because we are assuming (iii), the resulting open cover{Ox:x∈K}must have a finite subcover{Ox1, Ox2,. For each that is not compact, find an open cover for which there is no bounded subcover.
Perfect Sets and Connected Sets
Essentially, a set is connected if, no matter how it is split into two nonempty disjoint sets, it is always possible to show that at least one of the sets contains a limit point of the other. A andb ∈ B, no set is empty and, just as in Example 3.4.5 (ii), no set contains a boundary point of the other. We need to show that one of these sets contains a boundary point of the other.
Baire’s Theorem
A set B ⊆ R is called a Gδ set if it can be written as the countable intersection of open sets. This exercise already appeared as Exercise 3.2.15.) (a) Show that a closed interval [a, b] is aGδ set. b) Show that the half-open interval (a, b] is both aGδ and anFσ set. The set of real numbers R cannot be written as the countable union of nowhere-dense sets.
Epilogue
Extensions of the power of calculus were intimately linked to the ability to represent a function f(x) as a limit of polynomials (called a power series) or as a limit of sums of sines and cosines (called trigonometricorFourier- series). In this particular case, the definition of a functional limit on which we agree must lead to the conclusion that. However, we will quickly see that this topological formulation is equivalent to the sequential characterization we have arrived at here.
Functional Limits
Functional limits 117 that, for every ǫ-neighborhoodVǫ(L) of L, there exists a δ-neighborhoodVδ(c) around the property that for all different x∈Vδ(c) ngac (mex∈A) it follows that f(x) ∈ Vǫ(L). On a related note, there is no reason to discuss functional boundaries at isolated points in the domain. Thus, functional boundaries will only be considered as ascending to a boundary point of the domain of the function.
Continuous Functions
Assume:A→Rand g:A→Rare continuous in a pointc∈A. i) kf(x) is continuous atc for allk∈R;. iv) f(x)/g(x) is continuous at c, provided the quotient is defined. It is even simpler to show that a constant function f(x) = k, is continuous. It is sufficient to let δ = 1 regardless of the value ofǫ.) Because it is an arbitrary polynomial. Let f be a function defined on all of R, and assume there is a constant c such that 0< c < 1 and . a) Show that f is continuous on R.
Continuous Functions on Compact Sets
The function(x) = sin(1/x) (Fig.4.5) is continuous at every point of the open interval (0,1) but it is not uniformly continuous in this interval. In Example 4.4.3, we were unable to prove that g(x) = x2 is uniformly continuous on R because larger. Prove that g is uniformly continuous on (a, b) if and only if it is possible to determine the values g(a) and g(b) at the endpoints such that the extended function g is continuous on [a, b].
The Intermediate Value Theorem
Theorem 4.5.2 thus remains a valid conclusion in higher dimensions, while the intermediate value theorem is essentially a one-dimensional result. A typical way of applying the intermediate value theorem is to prove the existence of roots. Another way to summarize the intermediate value theorem is to say that every continuous function on [a, b] has the intermediate value property.
Sets of Discontinuity
Construct a bijection between the set of jump discontinuities of a monotone functionf and a subset of Q. Conclude that Df for a monotone function f must be either finite or countable but not uncountable. In Section 4.1 we constructed functions where the cutoff set was R (the Dirichlet function), R\{0} (the modified Dirichlet function), and Q (the Thomae function). a) Show that in each of the above cases we get a group anFσ as the set where the function is discontinuous. Given a function f onR, defineDαf to be the set of points where the function f fails to be α-continuous.
Epilogue
Given a functiong(x), the derivative′(x) is understood to be the slope of the graph ofg at any point in the domain. To summarize, the function g2(x) is continuous and differentiable everywhere in R (Fig.5.2), the derivative function g2′ is thus defined everywhere in R, but g′2 has an intercept at zero. We are looking for a function that is differentiable everywhere, including the zero point, where we are insisting that the slope of the graph be -1.
Derivatives and the Intermediate Value Property
This fact, together with the functional limit version of the Algebraic Limit Theorem (Theorem 4.2.4), justifies the conclusion. To find the correct formula for the derivative of the compound g◦f we can write. The function g2(x) discussed in Section 5.1 exhibits this behavior near c= 0.) The forthcoming proof of the chain rule manages to fine-tune this problem, but in substance it is essentially the argument just given.
The Mean Value Theorems
Combining this observation with the inner extreme theorem for differentiable functions (Theorem 5.2.6) gives rise to a special case of the mean value theorem, first noted by the mathematician Michel Rolle (Figure 5.5). It is the mean value theorem that gives us a way to rigorously articulate what appears to be geometrically valid. Use the generalized mean value theorem to provide a proof of the 0/0 case of L'Hospital's rule (Theorem 5.3.6).
A Continuous Nowhere-Differentiable Function
If we keep the first part of the hypothesis of Theorem 5.3.6 the same but assume that. Deciphering when results about finite sums of functions extend to infinite sums is one of the fundamental topics of Chapter 6. Putting xm=−(1/2m) in the previous argument produces difference coefficients that go to −∞. The geometric manifestation of this is the "cusp" that appears at x= 0 on the ofg graph.
Epilogue
Discussion: The Power of Power Series 171 Pluggingx= 1 into equations (1), (2), or (3) gives mathematical miserliness, so is it prudent to predict something meaningful coming out of equation (4) with the same value . Amidst all the unfounded assumptions we're making about infinity, calculations like this foster a sense of optimism about the legitimacy of our search for power series representations. Euler's idea was to continue factoring the power series in (6), and his strategy for doing so was very consistent with what we have seen so far—treat the power series as if it were a polynomial and after extending the model to infinity.
Uniform Convergence of a Sequence of Functions
Assume (fn) and (gn) are uniformly convergent series of functions. a) Show that (fn+gn) is a uniformly convergent sequence of functions. Uniform convergence of a sequence of functions 183 (a) Why does the sequence of real numbers fn(x1) necessarily contain a convergent subsequence (fnk). A sequence of functions (fn) defined on a setE⊆Ris called equicontinuous axes for everyǫ >0 there exists aδ >0 such that|fn(x)−fn(y)|< ǫ for alln∈Nand|x− y|< δ inE. a).
Uniform Convergence and Differentiation
Let (fn) be a series of differentiable functions defined on the closed interval [a, b], and assume that (fn′) converges uniformly on [a, b]. Let (fn) be a set of differentiable functions defined on the closed interval [a, b], and assume that (fn′) uniformly converges to a function gon [a, b]. Show that hn → 0 is uniform on R but that the series of derivatives (h′n) diverges for every x∈R. Show that this is differentiable in two ways:
Series of Functions
The advantages of uniform convergence over pointwise convergence suggest that a number of ways are needed to determine when a series converges uniformly.
Power Series
Taylor Series
The Weierstrass Approximation Theorem
Epilogue
The Definition of the Riemann Integral
Integrating Functions with Discontinuities
Properties of the Integral
The Fundamental Theorem of Calculus
Lebesgue’s Criterion for Riemann Integrability
Epilogue
Metric Spaces and the Baire Category Theorem
Euler’s Sum
Inventing the Factorial Function
Fourier Series
A Construction of R From Q
