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

Let f : A → B and g : B → C be functions.(a) Show that if g ο f is injective, then f is injective.(b) Show that if g ο f is surjective, then g is surjective.
Let f, g be functions such that (g ο f)(x) = x for all x ∈ D(f) and (f ο g)(y) = y for all y ∈ D (g). Prove that g = f-1.
If A and B are sets, show that A ⊂ B if and only if A ∩ B = A.
Prove the second De Morgan Law.
Prove the Distributive Laws: (a) A ∩ (B ⋃ C) = (A ∩ B) ⋃ (A ∩ C), (b) A ⋃ (B ∩ C) = (A ⋃ B) ∩ (A ⋃ C).
For each n ∊ N, let An = {(n + 1)}k : k ∈ N}. (a) What is A1 ∩ A2? (b) Determine the sets ⋃{An : n ∈ N} and ∩{An : n ∈ N}.
Prove that 1/1 ∙ 2 + 3 + ∙ ∙ ∙ + 1/n(n + 1) = n/(n + 1) for all n ∈ N.
Prove that 2n < n! for all n > 4, n ∈ N.
Prove that 2n 3 < 2n-2 for all n > 5, n ∈ N.
Find all natural numbers n such that n2 < 2n. Prove your assertion.
Find the largest natural number m such that n3 n is divisible by m for all n ∈ N. Prove your assertion.
Prove that 13 + 23 + ∙ ∙ ∙ + n3 = [1/2n(n + 1)2 for all n ∈ N.
Prove that 3 + 11 + ∙ ∙ ∙ + (8n 5) = 4n2 n for all n ∈ N.
Prove that 12 + 32 + ∙ ∙ ∙ + (2n - 1)2 = (4n3 - n)/3 for all n ∈ N.
Prove that 12 22 + 32 + ∙ ∙ ∙ + (-1)n+1n2 = (-1)n+1n(n + 1)/2 for all n ∈ N.
Prove that n3 + 5n is divisible by 6 for all n ∈ N.
Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n ∈ N.
Prove that a nonempty set T1 is finite if and only if there is a bijection from T1 onto a finite set T2.
(a) If (m, n) is the 6th point down the 9th diagonal of the array, calculate its number according to the counting method given.. (b) Given that h(m, 3) = 19, find m. For Information: for Theorem 1.3.8.
Determine the number of elements in P(S), the collection of all subsets of S, for each of the following sets: (a) S: = {1, 2}, (b) S: = {1, 2, 3}, (c) S: = {1, 2, 3, 4}. Be sure to include the empty set and the set S itself in P(S).
Use Mathematical Induction to prove that if the set S has n elements, then P(S) has 2n elements.
Prove that the collection F(N) of all finite subsets of N is countable.
Prove parts (b) and (c) of Theorem 1.3.4.
Let S : = {1, 2} and T := {a, b, c}. (a) Determine the number of different injections from S into T. (b) Determine the number of different surjections from T onto S.
Prove that a set T1 is denumerable if and only if there is a bijection from T1 onto a denumerable set T2.
Prove in detail that if S and T are denumerable, then S ∪ T is denumerable.
If a; b ∈ R, prove the following. (a) If a + b = 0, then b = -a, (b) -(-a) = a, (c) (-1)a = -a, (d) (-1 + (-1) = 1.
(a) If a < b and c < d, prove that a + c < b + d. (b) If 0 < a < b and 0 < c < d, prove that 0 < ac < bd.
(a) Show that if a > 0, then 1/a > 0 and 1/(1/a) = a. (b) Show that if a < b, then a < 1/2 (a + b) < b.
Let a, b, c, d be numbers satisfying 0 < a < b and c < d < 0. Give an example where ac < bd, and one where bd < ac.
If 0 < a < b, show that a2 < ab < b2. Show by example that it does not follow that a2 < ab < b2.
If 0 < a < b, show that (a) a < √ab < b, (b) 1/b < 1/a.
Find all real numbers x that satisfy the following inequalities. (a) x2 > 3x + 4; (b) 1 < x2 < 4; (c) 1/x < x; (d) 1/x < x2:
Prove that if a; b ∈ R, then(a) -(a + b) = (-a) + (-b),(b) (-a) ∙ (-b) = a ∙ b,(c) 1/(-a) = -(1/a),(d) -(a/b) = (-a)/b if b ≠ 0.
(a) If 0 < c < 1, show that 0 < c2 < c < 1. (b) If 1 < c, show that 1 < c < c2.
(a) Prove there is no n ∈ N such that 0 < n < 1. (Use the Well-Ordering Property of N.) (b) Prove that no natural number can be both even and odd.
(a) If c > 1, show that cn > c for all n ∈ N, and that cn > c for n > 1. (b) If 0 < c < 1, show that cn < c for all n ∈ N, and that cn < c for n > 1.
If a > 0; b > 0; and n ∈ N, show that a < b if and only if an < bn.
(a) If c > 1 and m; n ∈ N, show that cm > cn if and only if m > n. (b) If 0 < c < 1 and m; n ∈ N, show that cm < cn if and only if m > n.
Assuming the existence of roots, show that if c > 1, then c1/m < c1/n if and only if m > n.
Use Mathematical Induction to show that if a ∈ R and m; n ∈ N, then am+n = aman and (am) = amn.
Solve the following equations, justifying each step by referring to an appropriate property or theorem. (a) 2x + 5 = 8; (b) x2 = 2x; (c) x2 - 1 = 3; (d) (x - 1)(x + 2) = 0:
Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational number s such that s2 = 6.
Modify the proof of Theorem 2.1.4 to show that there does not exist a rational number t such that t2 = 3.
(a) Show that if x, y are rational numbers, then x + y and xy are rational numbers. (b) Prove that if x is a rational number and y is an irrational number, then x + y is an irrational number. If, in addition, x ≠ 0, then show that xy is an irrational number.
Let K := {s + t√2 : s; t ∈ Q}. Show that K satisfies the following: (a) If x1; x2 ∈ K, then x1 + x2 ∈ K and x1x2 ∈ K. (b) If x ≠ 0 and x ∈ K, then 1/x ∈ K. (Thus the set K is a subfield of R. With the order inherited from R, the set K is an ordered field that lies between Q and R.)
If a; b ∈ R and b ≠ 0, show that: (a) |a| = √a2; (b) |a/b| = |a|/|b|.
Find all x ∈ R that satisfy the following inequalities. (a) |x - 1| > |x + 1|; (b) |x| + |x + 1| < 2:
Find all x ∈ R that satisfy the inequality 4 < |x + 2| + |x - 1| < 5.
Find all x ∈ R that satisfy both |2x - 3| < 5 and |x + 1| > 2 simultaneously.
Determine and sketch the set of pairs (x; y) in R R that satisfy:(a) |x| = |y|;(b) |x| + |y| = 1;(c) |xy| + 2,(d) |x| - |y| = 2:
Determine and sketch the set of pairs (x, y) in R × R that satisfy: (a) |x| < |y|; (b) |x| + |y| < 1; (c) |xy| < 2; (d) |x| - |y| > 2:
Show that if a; b ∈ R then (a) max{a; b} = 1/2 (a + b + |a - b| and min{a; b} = 1/2 (a + b - |a - b|): (b) min{a; b; c} = min{min{a; b}; c}:
If x; y; z ∈ R and x < z, show that x < y < z if and only if |x - y| + |y - z| = |x - z|. Interpret this geometrically.
If a < x < b and a < y < b, show that |x - y| < b - a. Interpret this geometrically.
Find all x ∈ R that satisfy the following inequalities: (a) |4x - 5| < 13; (b) |x2 - 1| < 3:
Find all x ∈ R that satisfy the equation |x + 1| + |x - 2| = 7.
Find all values of x that satisfy the following equations: (a) x + 1 = |2x - 1|, (b) 2x - 1 = |x - 5|.
Find all values of x that satisfy the following inequalities. (a) |x - 2| < x + 1, (b) 3|x| < 2 - x:
Let S1: = {x ∈ R: x > 0}. Show in detail that the set S1 has lower bounds, but no upper bounds. Show that inf S1 = 0.
Show that if A and B are bounded subsets of R, then A ∪ B is a bounded set. Show that sup(A ∪ B) = sup{sup A; sup B}.
Let S ⊂ R and suppose that s* : = sup S belongs to S. If u ∉ S, show that sup(S ∪ {u}} = sup{s*; u}.
Show that a nonempty finite set S ⊂ R contains its supremum. [Use Mathematical Induction and the preceding exercise.]
Let S be a set that is bounded below. Prove that a lower bound w of S is the infimum of S if and only if for any ε > 0 there exists t ∈ S such that t < w + ε.
Let S2 :¼ {x ∈ R : x > 0}. Does S2 have lower bounds? Does S2 have upper bounds? Does inf S2 exist? Does sup S2 exist? Prove your statements.
Find the infimum and supremum, if they exist, of each of the following sets. (a) A: = {x ∈ R : 2x + 5 > 0}; (b) B: = {x ∈ R : x + 2 > x2} (c) C: = {x ∈ R : x < 1/x}; (d) D: = {x ∈ R : x2 - 2x - 5 < 0}
Let S be a nonempty subset o{ R that is bounded below. Prove that inf S = {sup{-s: s ∈ S}.
Let S ⊂ R be nonempty. Show that if u = sup S, then {or every number n ∈ N the number u - 1/n is not an upper bound of S, but the number u + 1/n is an upper bound of S. (The converse is also true.
Show that sup{1 - 1/n : n ∈ N} = 1.
Let X and Y be nonempty sets and let h : X Ã— Y †’ R have bounded range in R. Let f : X †’ R and g : Y †’ R be defined byF(x) := sup{h(x; y) : y ˆˆ Y}; g(y) := inf{h(x; y) : x ˆˆ X}:Prove thatSup{g(y) : y ˆˆ Y} We sometimes express this by writingExercises 9 and 10 show that
Let X and Y be nonempty sets and let h : X Ã— Y †’ R have bounded range in R. Let F : X †’ R and G : Y †’ R be defined byF(x) := sup{h(x; y) : y ˆˆ Y}; G(y) := sup{h(x; y) : x ˆˆ X}:Establish the Principle of the Iterated
Given any x ∈ R, show that there exists a unique n ∈ Z such that n - 1 < x < n.
Modify the argument in Theorem 2.4.7 to show that there exists a positive real number y such that y2 = 3.
Modify the argument in Theorem 2.4.7 to show that if a > 0, then there exists a positive real number z such that z2 = a.
Modify the argument in Theorem 2.4.7 to show that there exists a positive real number u such that u3 = 2.
If S := {1/n - 1/m : n;m ∈ N}, find inf S and sup S.
Let S ⊂ R be nonempty. Prove that if a number u in R has the properties: (i) for every n ∈ N the number u - 1/n is not an upper bound of S, and (ii) for every number n ∈ N the number u + 1/n is an upper bound of S, then u = sup S. (This is the converse of Exercise 2.3.9.)
Let S be a nonempty bounded set in R. (a) Let a > 0, and let aS := {as : s ∈ S}. Prove that Inf(aS) = a inf S; sup(aS) = a sup S: (b) Let b < 0 and let bS = {bs : s ∈ S}. Prove that Inf(bS) = b sup S; sup(bS) = b inf S:
Let S be a set of nonnegative real numbers that is bounded above and let T := {x2 : x ∈ S}. Prove that if u = sup S, then u2 = sup T. Give an example that shows the conclusion may be false if the restriction against negative numbers is removed.
Let X be a nonempty set and let f : X → R have bounded range in R. If a ∈ R, show that Example 2.4.l(a) implies that Sup{a + f(x) : x ∈ X} = a + sup{f(x) : x ∈ X}: Show that we also have Inf{a + f(x) : x ∈ X} = a + inf{f(x) : x ∈ X}:
Let A and B be bounded nonempty subsets of R, and let A þ B :¼ fa þ b : a 2 A; b 2 Bg. Prove that sup(A + B) = sup A + sup B and inf(A + B) = inf A + inf B.
Let X be a nonempty set, and let f and g be defined on X and have bounded ranges in R. Show that Sup{f(x) + g(x) : x ∈ X} < supf{x) : x ∈ X} + sup{g(x) : x ∈ X} and that inf{f(x) : x ∈ X} + inf{g(x) : x ∈ X} < inf{f(x) + g(x) : x ∈ X}: Give examples to show that each of these
With the notation in the proofs of Theorems 2.5.2 and 2.5.3, show that we have Î· ˆˆ ˆ©ˆžn=1In. Also show that
Show that the intervals obtained from the inequalities in (2) form a nested sequence.
Give the two binary representations of 3/8 and 7/16.
Show that if ak; bk ∈ {0; 1; . . . ; 9} and if a1/10 + a2/102 +∙ ∙ ∙+ an/10n = b1/10 + b2/102+∙ ∙ ∙+ bm/10m ≠ 0; then n = m and ak = bk for k = 1; . . . ; n.
Find the decimal representation of -2/7.
If S ⊂ R is a nonempty bounded set, and IS := (inf S; sup S) , show that S ⊂ IS. Moreover, if J is any closed bounded interval containing S, show that IS ⊂ J.
Write out the details of the proof of Case (iv) in Theorem 2.5.1.
If I1 ⊂ I2 ⊃ ∙ ∙ ∙⊃ In is a nested sequence of intervals and if In = [an; bn], show that a1 < a2 < ∙ ∙ ∙ < an < ∙ ∙ ∙ and b1 > b2 > ∙ ∙ ∙ bn > ∙ ∙ ∙.
Let In := [0; 1/n] for n ∈ N. Prove that ∩∞n=1In = {0}.
Let x1 := 1 and xn+1 := √2 + xn for n ∈ N. Show that (xn) converges and find the limit.
Let y1 := √p, where p > 0, and yn+1 := √p + yn for n ∈ N. Show that (yn) converges and find the limit. [One upper bound is 1 + 2√p.]
Let a > 0 and let z1 > 0: Define zn+1 := √a + zn for n ∈ N. Show that (zn) converges and find the limit.
Let x1 : = a > 0 and xn+1 : = xn + 1/xn for n ∈ N. Determine whether (xn) converges or diverges.
Let (an) be an increasing sequence, (bn) be a decreasing sequence, and assume that an < bn for all n ∈ N. Show that lim(an) < lim(bn), and thereby deduce the Nested Intervals Property 2.5.2 from the Monotone Convergence Theorem 3.3.2.
Let (xn) be a bounded sequence and for each n ∈ N let sn := sup{xk : k > n} and S := inf{sn}. Show that there exists a subsequence of (xn) that converges to S.
Let (In) be a nested sequence of closed bounded intervals. For each n ∈ N, let xn ∈ In. Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals Property 2.5.2.
Show that if (xn) is a bounded sequence, then (xn) converges if and only if lim sup(xn) = lim inf(xn).
Show that if (xn) and (yn) are bounded sequences, then lim sup(xn + yn) < lim sup(xn) + lim sup(yn): Give an example in which the two sides are not equal.

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