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Introduction to Real Analysis 4th edition Robert G. Bartle, Donald R. Sherbert - Solutions

Suppose (fn) is a sequence of continuous functions on an interval I that converges uniformly on I to a function f. If (xn) ⊂ I converges to x0 ∈ I, show that lim(fn(xn)) = f (x0).
Let fn(x) := 1/(1 + x)n for x ∈ [0, 1]. Find the point wise limit f of the sequence (fn) on [0, 1]. Does (fn) converge uniformly to f on [0, 1]?
Suppose the sequence (fn) converges uniformly to f on the set A, and suppose that each fn is bounded on A. (That is, for each n there is a constant Mn such that | fn(x)| < Mn for all x ∈ A.) Show that the function f is bounded on A.
Let fn(x) := nx/(1 + nx2) for x ∈ A := [0, ∞). Show that each fn is bounded on A, but the point wise limit f of the sequence is not bounded on A. Does (fn) converge uniformly to f on A?
Let fn (x) := xn/n for x ∈ [0, 1]. Show that the sequence (fn) of differentiable functions converges uniformly to a differentiable function f on [0, 1], and that the sequence (fʹn) converges on [0, 1] to a function g, but that g(1) ≠ fʹ(1).
Show that if x > 0 and if n > 2x, thenUse this formula to show that 2 2/3
Evaluate Lʹ(1) by using the sequence (1 + 1/n) and the fact that e = lim(1 + 1/n)n).
(a) Show that if a > 0, then the function x †’ xa is strictly increasing on (0, ˆž) to R and that(b) Show that if a
Prove that if a > 0, a ≠ 1, then alogax = x for all x ∈ (0, ∞) and loga(ay) = y for all y ∈ R. Therefore the function x → logax on (0, →) to R is inverse to the function y → ay on R.
Show that if 0
Show that if n > 2, thenUse this inequality to prove that e is not a rational number.
Let f : R → R be such that fʹ(x) = f(x) for all x ∈ R. Show that there exists K ∈ R such that f(x) = Kex for all x ∈ R.
Calculate cos(.2), sin(.2) and cos 1, sin 1 correct to four decimal places.
Show that c(x) > 1 for all x ˆˆ R, that both c and s are strictly increasing on (0, ˆž), and that
Show that property (vii) of Theorem 8.4.8 does not hold if x < 0, but that we have |sin x| < |x| for all x ∈ R. Also show that |sin x - x| < |x|3/6 for all x ∈ R.
Calculate π by approximating the smallest positive zero of sin. (Either bisect intervals or use Newton's Method of Section 6.4.)
Define the sequence (cn) and (sn) inductively by c1(x) := 1, s1(x):= x, andfor all n ˆˆ N, x ˆˆ R. Reason as in the proof of Theorem 8.4.1 to conclude that there exist functions c : R †’ R and s : R †’ R such that (j) cÊ¹Ê¹(x) = c(x) and
If f : R → R is such that fʹʹ(x) = f(x) for all x ∈ R, show that there exist real numbers a, b such that f(x) = ac(x) + bs(x) for all x ∈ R. Apply this to the functions f1(x) := ex and f2(x) := e-x for x ∈ R. Show that c(x) = 1/2 (ex + e-x) and s(x) = 1/2 (ex - e-x) for x ∈ R.
Show that if a convergent series contains only a finite number of negative terms, then it is absolutely convergent.
Let a > 0. Show that the series ∑(1 + an)-1 is divergent if 0 < a < 1 and is convergent if a > 1.
(a) Does the series(b) Does the series
If (ank) is a subsequence of (an), then the series ∑ ank is called a subseries of ∑ an. Show that ∑ an is absolutely convergent if and only if every subseries of it is convergent.
Suppose aij > 0 for I, j ˆˆ N. If (ck) is any enumeration of {aij : i, j ˆˆ N}, show that the following statements are equivalent:
The preceding exercise may fail if the terms are not positive. For example, let aij := +1 if i - j = 1, aij := -1 if i - j = -1, and aij := 0 elsewhere. Show that the iterated sumsBoth exist but are not equal.
Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.
If ∑an is absolutely convergent, is it true that every rearrangement of ∑an is also absolutely convergent?
(a) If ∑an is absolutely convergent and (bn) is a bounded sequence, show that ∑anbn is absolutely convergent. (b) Give an example to show that if the convergence of ∑an is conditional and (bn) is a bounded sequence, then ∑anbn may diverge.
If (an) is a decreasing sequence of strictly positive numbers and if ∑an is convergent, show that lim(nan) = 0.
Establish the convergence or the divergence of the series whose nth term is: (a) 1/(n + 1)(n + 2), (b) n/(n + 1)(n + 2), (c) 2-1/n, (d) n/2n.
If r ∈ (0, 1) satisfies (5) in the Ratio Test 9.2.4, show that |s - sn| < r|xn|/(1 - r) for n > K.
If a > 1 satisfies (10) in Raabe's Test 9.2.8, show that |s - sn| < n|xn|/(a - 1) for n > K.
For n ˆˆ N, let cn be defined by cn := 1/1 + 1/2 + ˆ™ ˆ™ ˆ™+1/n - ln n. Show that (cn) is a decreasing sequence of positive numbers. Show that if we putthen the sequence (bn) converges to ln 2. [bn = c2n - cn + ln 2.]
Let {n1, n2, . . .} denote the collection of natural numbers that do not use the digit 6 in their decimal expansion. Show that ∑1/nk converges to a number less than 80. If {m1, m2, . . .} is the collection of numbers that end in 6, then ∑1/mk diverges. If {p1, p2, . . . } is the collection of
If p > 0, q > 0, show that the seriesconverges for q > p + 1 and diverges for q
Suppose that none of the numbers a, b, c is a negative integer or zero. Prove that the hypergeometric seriesis absolutely convergent for c > a + b and divergent for c
Let an > 0 and suppose that ∑an converges. Construct a convergent series ∑ bn with bn > 0 such that lim(an/bn)= 0; hence ∑bn converges less rapidly than ∑an. [Let (An) be the partial sums of ∑an and A its limit. Define b1 := √A - √A - A1 and bn := √A - An-1 - √A - An for n > 1.]
Establish the convergence or divergence of the series whose nth term is: (a) (n(n + 1)-1/2. (b) (n2(n + 1))-1/2, (c) n!/nn, (d) (-1)nn/(n + 1).
Let (an) be a decreasing sequence of real numbers converging to 0 and suppose that ∑an diverges. Construct a divergent series ∑bn with bn > 0 such that lim(bn/an)= 0; hence ∑bn diverges less rapidly than ∑an. [Let bn := an/√An where An is the nth partial sum of ∑an.]
Discuss the convergence or the divergence of the series with nth term (for sufficiently large n) given by (a) (ln n)-p, (b) (ln n)-n, (c) (ln n)-ln n, (d) (ln n)-ln ln n (e) (n ln n)-1, (f) (n(ln n)(ln ln n)2)-1
Discuss the convergence or the divergence of the series with nth term: (a) 2ne-n, (b) nne-n, (c) e-ln n, (d) (ln n)e-√n, (e) n!e-n, (f) n!e-n2.
Discuss the series whose nth term is(a)(b) (c) (d)
Test the following series for convergence and for absolute convergence:(a)(b) (c) (d)
If the partial sums sn of are bounded, show that the series converges to
Can Dirichlet's Test be applied to establish the convergence ofwhere the number of signs increases by one in each ''block''? If not, use another method to establish the convergence of this series.
Show that the hypothesis that the sequence X := (xn) is decreasing in Dirichlet's Test 9.3.4 can be replaced by the hypothesis that is convergent.
If (an) is a bounded decreasing sequence and (bn) is a bounded increasing sequence and if xn := an + bn for n ˆˆ N, show that is convergent.
Show that if the partial sums sn of the series satisfy |sn| converges.
Suppose that ∑ an is a convergent series of real numbers. Either prove that ∑ bn converges or give a counter-example, when we define bn by (a) an/n. (b) √an/n (an > 0), (c) an sin n. (d) √an/n (an > 0), (e) n1/nan, (f) an/(1 + |an|).
If sn is the nth partial sum of the alternating series and if s denotes the sum of this series, show that |s - sn|
Consider the serieswhere the signs come in pairs. Does it converge?
(a) Show that 0
Let M(x) := ln|x| for x ≠ 0 and M(0) := 0. Show that Mʹ(x) = 1/x for all x ≠ 0. Explain why it does not follow that ∫2-2 (1/x)dx = ln|-2| - ln 2 = 0.
Let E := {c1, c2, . . .} and let F be continuous on [a, b] and FÊ¹(x) = f(x) for x ˆˆ [a, b]E and f(ck) := 0. We want to show that f ˆˆ R*[a, b] and that equation (5) holds.(a) Given Îµ > 0 and t ˆˆ [a, b]E, let Î´Îµ(t) be
Show that the function g1(x) := x-1/2sin(1/x) for x ∈ [0, 1] and g1(0) := 0 belongs to R*[0, 1]. [Differentiate c1(x) := x3/2cos(1/x) for x ∈ [0, 1] and c1(0) := 0.]
Use the Substitution Theorem 10.1.12 to evaluate the following integrals:(a)(b)(c)(d)
Let F(x) := x cos(Ï€/x) for x ˆˆ [0; 1] and F(0) := 0. It will be seen that f := FÊ¹ ˆˆ R*[0, 1] but that its absolute value |f| = |FÊ¹| ˆ‰ R*[0, 1]. (Here f(0) := 0.)(a) Show that FÊ¹ and |FÊ¹| are continuous on
(b) Are there tagged partitions in which every tag belongs to exactly two subintervals?
Let f be as in Exercise 19 and let m(x) := (-1)k for x ∈ [ak, bk](k ∈ N), and m(x) := 0 elsewhere in [0, 1]. Show that m ∙ f = |m ∙ f|. Use Exercise 7.2.11 to show that the bounded functions m and |m| belong to R[0, 1]. Conclude that the product of a function in R*[0, 1] and a bounded
Let Φ(x) := x|cos(π/x)| for x ∈ [0, 1] and let Φ(0+ := 0. Then Φ is continuous on [0, 1] and Φʹ(x) exists for x ∉ E1 := {0} ⋃ {ak : k ∈ N}, where ak := 2/(2k + 1). Let ω(x) := Φʹ(x) for x ∉ E and ω(x) := 0 for x ∈ E. Show that w is not bounded on [0,1]. Using the Fundamental
Let Ψ(x) := x2|cos(p/x)| for x ∈ [0, 1] and Ψ(0) := 0. Then Ψ is continuous on [0, 1] and Ψʹ(x) exists for x ∉ E1 := {ak}. Let Ψ(x) := Ψʹ(x) for x ∉ E1, and ψ(x) := 0 for x ∈ E1. Show that ψ is bounded on [0, 1] and (using Exercise 7.2.11) that ψ ∈ R[0, 1]. Show that ∫ac ψ =
If f : [a, b] †’ R is continuous and if p ˆˆ R*[a, b] does not change sign on [a, b], and if f p ˆˆ R*[a, b], then there exists (This is a generalization of Exercise 7.2.16, it is called the First Mean Value Theorem for integrals.)
Let f ˆˆ R*[a, b], let g be monotone on [a, b] and suppose that f > 0. Then there exists such that (This is a form of the Second Mean Value Theorem for integrals.)
Let Î´ be a gauge on [a, b] and let be a Î´-fine partition of [a, b].(a) Show that there exists a Î´-fine partition 1 such that (i) no tag belongs to two subintervals in 1, and (ii) S(f; 1) = S(f; ) for any function f on [a, b].(b) Does there exist a Î´-fine partition 2 such that (j) every
If Î´ is defined on [0, 2] by Î´(t) := 1/2 |t - 1| for x ‰ 1 and Î´(1) := 0.01, show that every Î´-fine partition of [0, 2] has t = 1 as a tag for at least one subinterval, and that the total length of the subintervals in having 1 as a
(a) Construct a gauge δ on [0, 4] that will force the numbers 1, 2, 3 to be tags of any d-fine partition of this interval. (b) Given a gauge δ1 on [0, 4], construct a gauge δ2 such that every δ2-fine partition of [0, 4] will (i) have the numbers 1, 2, 3 in its collection of tags, and (ii) be
Show that f ˆˆ R*[a, b] with integral L if and only if for every Îµ > 0 there exists a gauge ge on [a, b] such that if = {([xi-1, xi), ti)}ni=1 is any tagged partition such that 0 Î³Îµ(ti) for i = 1, . . . , n, then |S(f, P) - L|
Show that the following functions belong to R*[0, 1] by finding a function Fk that is continuous on [0, 1] and such that Fʹk(x) = fk(x) for x ∈ [0, 1]\Ek, for some finite set Ek. (a) f1(x) := (x + 1)/√x for x ∈ [0, 1] and f1(0) := 0. (b) f2(x) := x/√1 - x for x ∈ [0, 1] and f2(1) :=
Explain why the argument in Theorem 7.1.6 does not apply to show that a function inR*[a, b] is bounded.
(a) Give an example of a function f ∈ R* [0, 1] such that max {f, 0} does not belong to R*[0; 1]. (b) Can you give an example of f ∈ L[0, 1] such that max{f, 0} ∉ {2 L[0, 1]?
Write out the details of the proof that min{f, g} ∈ R*[a, b] in Theorem 10.2.8 when a < f and a < g.
Write out the details of the proofs of Theorem 10.2.11.
If f, g ∈ L[a, b], show that | ||f|| | ||g|| < ||f ± g||.
Establish the easy part of the Completeness Theorem 10.2.12
(a) Apply Hake's Theorem to conclude that g(x) := 1/x2/3 for x ∈ [0, 1] and g(0) := 0 belongs to R*(0, 1). (b) Explain why Hake's Theorem does not apply to f(x) := 1/x for x ∈ (0, 1) and f(0) := 0 (which does not belong to R*[0, 1]).
Suppose that f ∈ R* = [a, c] for all c ∈ (a, b) and that there exists γ ∈ (a, b) and ω ∈ L[a, b] such that |f(x)| < v(x) for x ∈ [γ, b]. Show that f ∈ R* = [a, b].
Show that the function g1(x) := x-1/2sin(1/x) for x ∈ (0, 1) and g1(0) := 0 belongs to L[0, 1]. (This function was also considered in Exercise 10.1.15.)
Show that the following functions (properly defined when necessary) are in L[0, 1]. (a) x ln x/1 + x2, (b) sin π x/ln x, (c) (ln x)(ln(1 - x)), (d) In x/√1 - x2.
Determine whether the following integrals are convergent or divergent. (Define the integrands to be 0 where they are not already defined.)(a)(b) (c) (d) (e) (f)
(a) Use Integration by Parts and Hake's Theorem to show that ∫∞0 xe-sxdx = 1/s2 for s > 0. (b) Use the Fundamental Theorem 10.3.5.
(a) Show that the integral ∫∞1 x-1 ln x dx does not converge. (b) Show that if a > 1, then ∫∞1 x-a ln x dx = 1/(a - 1)2.
(a) Show that ∫(n+1)πnπ |x-1sin x|dx > ¼(n + 1): (b) Show that |D| ∉ R*[0, 1], where D is as in Example 10.3.4(d).
Show that the integral ∫∞0(1/√x) sin x dx converges. [Integrate by Parts.]
Establish the convergence or the divergence of the following integrals:(a)(b) (c) (d) (e) (f)
Let f, φ : [a, ∞] → R. Abel's Test asserts that if f ∈ R*[a, ∞] and φ is bounded and monotone on [a, ∞], then f φ ∈ R*[a, ∞].(a) Show that Abel's Test does not apply to establish the convergence of ∫∞0 (1/x)sin x dx by taking φ(x) := 1/x. However, it does apply if we take
(a) Show that the integral ˆ«ˆž0(1/x) sin x dx converges.(b) Show that ˆ«ˆž2 (1/ln xÃž sin x dx converges.(c) Show that ˆ«ˆž0 (1/ˆšx) cos x dx converges.(d) Show that the Chartier-Dirichlet Test does not apply to establish the convergence of ˆ«ˆž0(x/(x + 1)) sin(x2)dx by taking
Let f ∈ R*[a, g] for all γ > a. Show that f ∈ R*[a, ∞] if and only if for every ε > 0 there exists K(ε) > a such that if q > p > K(ε), then |∫qp f| < ε.
Establish the convergence of the following integrals:(a)(b) (c) (d)
Let f and |f| belong to R*[a, γ] for all γ > a. Show that f ∈ L[a, ∞] if and only if for every ε > 0 there exists K(ε) > a such that if q > p > K(ε) then ∫qp |f| < δ.
Let f and |f| belong to R*[a, γ] for every γ > Show that F ∈ [a, ∞] if and only if the set V: = {∫xa |f| : x > a} is bounded in R.
If f, g ∈ L[a, ∞], show that f ± g ∈ L[a,1]. Moreover, if ||h|| :∫∞a |h| for any h ∈ L[a,∞], show that ||f + g|| < ||f|| + ||g||.
If f(x) := 1/x for x ∈ [1,∞}, show that f ∉ R*[1,1].
If f is continuous on [1, ∞] and if |f(x)| < K/x2 for x ∈ [1, ∞], show that f ∈ L[1,∞].
If s > 0, let g(x) := e-sx for x ∈ [0, ∞]. (a) Use Hake's Theorem to show that g ∈ L[0, ∞] and ∫10 e-sxdx = 1/s. (b) Use the Fundamental Theorem 10.3.5.
Consider the following sequences of functions with the indicated domains. Does the sequence converge? If so, to what? Is the convergence uniform? Is it bounded? If not bounded, is it dominated? Is it monotone? Evaluate the limit of the sequence of integrals. (a) kx/1 + kx [0, 1]. (b) xk/1 + xk [0,
If t > 0, define E(t) := ˆ«ˆž0 [(e-txsin x)/x]dx .(a) Show that E exists and is continuous for t > a > 0. Moreover, E(t) †’ 0 as t †’ ˆž.(c) Deduce that E(t) = 1/2 Ï€ - Arctan t for t > 0. (d) Explain why we cannot use the formula in (c)
In this exercise we will establish the important formula:(a) Let G(t) := ˆ«10 [e-t2(x2+1)/(x2 + 1)]dx for t > 0. Since the integrand is dominated by 1/(x2 + 1) for t > 0, then G is continuous on [0, ˆž). Moreover, G(0) = Arctan 1 = 1/4 Ï€ and it follows from the
Suppose I Š‚ R is a closed interval and that f : [a; b] Ã— I †’ R is such that exists on [a, b] Ã— I, and for each t ˆˆ [a, b] the function x †’ f(t, x) is in R*(I) and there exist a, Ï‰ ˆˆ R*(I) such that the
(a) If f, g ∈ M[a, b], show that max{f, g} and min{f, g} belong to M[a, b]. (b) If f, g, h ∈ M[a, b], show that mid{f, g, h} ∈ M[a, b].
(a) If (fk) is a bounded sequence in M[a, b] and fk → f a.e., show that f ∈ M[a, b]. [Use the Dominated Convergence Theorem.](b) If (gk) is any sequence in M[a, b] and if fk := Arctan ο gk, show that (fk) is a bounded sequence in M[a, b].(c) If (gk) is a sequence in M[a, b] and if gk → g
A set E in [a, b] is said to be (Lebesgue) measurable if its characteristic function 1E (defined by 1ε(x) := 1 if x ∈ E and 1ε(x) := 0 if x ∈ [a, b]\E) belongs to M[a, b]. We will denote the collection of measurable sets in [a, b] by M [a, b]. In this exercise, we develop a number of

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