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An Introduction to Analysis 4th edition William R. Wade - Solutions

Prove that if xn-1, xn satisfy (19), then xn is the x-intercept of the tangent line to y = f(x) at the point (xn-1, f(xn-1)).
Prove that cos(l) is irrational.
Suppose that f : R → R. If f" exists and is bounded on R, and there is an ε0 > 0 such that |f'(x)| > ε0 for all x ∈ R, prove that there exists a δ > 0 such that if |f(x0)| < δ > 0 for some x0 ∈ R, then f has a root; that is, that f(c) = 0 for some c ∈ R.
Let x ˆˆ [0, 1) and an, Î²n be defined as in Theorem 7.60.a) If f : [0, 1) †’ R and Î³ ˆˆ R, prove thatb) If f is differentiable at x, prove that (26) holds.
Let x, y, z ∈ Rʹʹ a) If ||x - z|| < 2 and ||y - z|| < 3, prove that ||x - y|| < 5. b) If ||x|| < 2, ||y|| < 3, and ||z|| < 4, prove that |x ∙ y - x ∙ z| < 14. c) If ||x - y|| < 2 and ||z|| < 3, prove that |x ∙ (y - z) - y ∙ (x - z)| < 6. d) If ||2x - y|| < 2 and ||y|| < 1, prove that |
Prove that the 1-norm and the sup-norm also satisfy Theorem 8.6.
Let B :={x ˆˆ Rn : ||x|| a) If a, b, c ˆˆ B andprove that v belongs to B. b) If a, b ˆˆ B, prove that for all c, d ˆˆ Rn. c) If a, b, c ˆˆ B and n = 3, prove that
Use the proof of Theorem 8.5 to show that equality in the Cauchy-Schwarz Inequality holds if and only if x = 0, y = 0, or x is parallel to y.
Let a and b be nonzero vectors in Rn. a) If ɸ(t) = a + tb for t ∈ R, show that for each to, t0, t1, t2 ∈ R with t1, t2 ≠ t0, the angle between ɸ(t0) - ɸ(t0) - ɸ(t0) and ɸ(t2) - ɸ(t0) is 0 or π. b) If θ is the angle between a and b, show that a and b are parallel according to
The midpoint of a side of a triangle in R3 is the point that bisects that side (i.e., that divides it into two equal pieces). Let Δ be a triangle in R3 with sides A, B, and C and let L denote the line segment between the midpoints of A and B. Prove that L is parallel to C and that the length of L
a) Prove that (1,2,3), (4,5,6), and (0,4,2) are vertices of a right triangle in R3.b) Find all nonzero vectors orthogonal to (1,-1, 0) which lie in the plane z = x.c) Find all nonzero vectors orthogonal to the vector (3, 2, -5) whose components sum to 4.
Let a < b be real numbers. The Cartesian product [a, b] × [a, b] is obviously a square in R2. Define a cube Q in R" to be the n-fold Cartesian product of [a, b] with itself; that is, Q := [a, b] × ∙ ∙ ∙ × [a, b]. Find a formula of the angle between the longest diagonal of Q and any of its
a) Using Postulate 1 in Section 1.2 and Definition 8.1, prove Theorem 8.2.b) Prove Theorem 8.9, parts i) through iii) and vi).c) Prove that if x, y ∈ R3, then ||x × y|| < ||x|| ||y||.
Suppose that {(ak) and are sequences of real numbers which satisfyProve that the infinite series ˆ‘ˆžk=1 akbk converges absolutely.
Let a, b, c ∈ R3.a) Prove that if a, b, and c do not all lie on the same line, then an equation of the plane through these points is given by (x, y, z) • d = a • d, whered := (a - b) × (a - c).b) Prove that if c does not lie on the line ɸ(t) = ta + b, ∈ R, then an equation of the plane
For each of the following functions f, find the matrix representative of a linear transformation T ˆˆ £(R; R'") which satisfiesa) f(x) = (x2, sinx)- b) f(x) = (ex, 3ˆšx, l - x2) c) f(x) = (1, 2, 3, x2 + x, x2 -x)
Fix T ˆˆ £(R"; RW). Seta) Prove that M1 b) Using the linear property of T, prove that if x ‰ 0, then c) Prove that M1 = M2 = ||T||.
a) Find an equation of the hyperplane through the points (1, 0, 0, 0), (2,1, 0, 0), (0,1,1, 0), and (0, 4, 0,1).b) Find an equation of the hyperplane that contains the lines ɸ(t) = (t, t, t, 1) and ψ(t) = (1, t, 1 + t, t), t ∈ R.c) Find an equation of the plane parallel to the hyperplane x1 +
Find two lines in R3 which are not parallel but do not intersect.
Suppose that T ∈ £(R"; R'") for some n, m ∈ N.a) Find the matrix representative of T if T(x, y, z, w) = (0, x + y, x - z, x + y + w).b) Find the matrix representative of T if T(x, y, z) = x - y + z.c) Find the matrix representative of T if T(x1, x2,..., xn) = (x1 - xn, xn - x1).
Suppose that T ∈ £(R"; R'") for some n,m ∈ N. a) If T(l, 1) = (3, π, 0) and T(0, 1) = (4,0, 1), find the matrix representative of T. b) If T(l, I, 0) = (e, π), T(0,-1, 1) = (1, 0), and T(l, 1,-1) = (1.2), find the matrix representative of T. c) If T(0, 1, 1, 0) = (3, 5), T(0, 1, -1, 0) = (5,
Suppose that a, b, c ˆˆ R3 are three points which do not lie on the same straight line and that U is the plane which contains the points a, b, c. Prove that an equation of II is given by
Recall that the area of a parallelogram with base b and altitude h is given by bh, and the volume of a parallelepiped is given by the area of its base times its altitude. a) Let a, b ∈ R3 be nonzero vectors and V represent the parallelogram {(x, y, z) = ua + vb : u, v ∈ [0, 1]}. Prove that the
The distance from a point x0 = (x0, y0, z0) to a plane II in R3 is defined to bewhere v := (x0 - x1, y0 - y1, z0 - z1) for some (x1, y1, z1) ˆˆ II, and v is orthogonal to II (i.e., parallel to its normal). Sketch II and xo for a typical plane II, and convince yourself that this is the
a) Prove that ||B(x, y)|| = ||(x, y)|| for all (x, y) ˆˆ R2.b) Let (x, y) ˆˆ R2 be a nonzero vector and Ï† represent the angle between B(x, y) and (x, y). Prove that cos Ï† = cos Î¸. Thus, show that B rotates R2 through an angle Î¸. (When Î¸ > 0, we shall call B counterclockwise
Sketch each of the following sets. Identify which of the following sets are open, which are closed, and which are neither. Also discuss the connectivity of each set.a) E = {(x, y): y ≠ 0}b) E = {(x, y): x2 + 4y2 < 1}c) E = {(x, y): y > x2, 0 < y < 1}d) E = \(x, y): x2 - y2 > 1, -1
Graph generic open balls in R2 with respect to each of the "non-Euclidean" norms || ∙ ||1 and || ∙ ||∞. What shape are they?
Let n ∈ N, let a ∈ R", let s, r ∈ R with s < r, and setV = {x ∈ R": s < ||x - a|| < r} and E = {x ∈ R" : s < ||x - a|| < r}.Prove that V is open and E is closed.
a) Let a < b and c < d be real numbers. Sketch a graph of the rectangle[a, b] × [c, d] := {(x, y) : x ∈ [a, b], y ∈ [c, d]},and decide whether this set is connected. Explain your answers,b) Sketch a graph of setβ1(-2, 0) U β1(2, 0) U {(x, 0) : -1 < x < 1},and decide whether this
a) Let E1 denote the closed ball centered at (0, 0) of radius 1 and E2 := β√2(2, 0), and sketch a graph of the set U: = {(x, y): x2 + y2 < 1 and x2 - 4x + y2 + 2 < 0}. b) Decide whether U is relatively open or relatively closed in E1. Explain your answer. c) Decide whether U is relatively open
Suppose that E ⊂ R" and that C is a subset of E.a) Prove that if E is closed, then C is relatively closed in E if and only if C is (plain old vanilla) closed (in the usual sense).b) Prove that C is relatively closed in E if and only if E\C is relatively open in E.
a) If A and B are connected in R" and A ˆ© B ‰ 0, prove that A U B is connected.b) If {Ea}aˆˆA is a collection of connected sets in R" and ˆ©aˆˆAEa ‰ 0, prove thatis connected.c) If A and B are connected in R and A ˆ© B ‰ Î¸, prove that A ˆ© B is connected.d)
Let V be a subset of R".a) Prove that V is open if and only if there is a collection of open balls {BÎ± : Î± ˆˆ A} such thatb) What happens to this result when open is replaced by closed?
Show that if E is closed in R" and a ˆ‰ E, then
Find the interior, closure, and boundary of each of the following subsets of R.a) E = {l/n: n ˆˆ N|b)c) E = Uˆžn=1(-n, n)d) E = Q
Let A and B be subsets of Rn.a) Show that ϑ{A ∩ B) ∩ (AC U (ϑB)C) ⊂ 3A.b) Show that if x ∈ ϑ(A ∩ B) and x ∉ (A ∩ ϑB)U(B ∩ ϑA), then x ∈ ϑA∩ϑB.c) Prove that ϑ{A ∩ B) ⊂ (A ∩ ϑB) U (B ∩ ϑA) U (ϑA ∩ ϑB).d) Show that even in R, there exist sets A and B such that
Let E ⊂ Rn and U be relatively open in E. a) If ⋃ ⊂ E°, then ⋃ ∩ ϑ⋃ = θ. b) If U ∩ ϑE ≠ 0, then U ∩ ϑU = ⋃ ∩ ϑE.
For each of the following sets, sketch E°, , and ϑE.a) E = {(x, y) : x2 + 4y2 < 1}b) E = {(x, y) : x2 - 2x + y2 = 0} U {(x, 0) : x ∈ [2, 3]}c) E = {(x, y) : y > x2, 0 < y < 1}d) E = {(x, y) : x2 - y2 < 1, -1 < y < 1}
Let E be a subset of R". a) Prove that every subset A ⊂ E contains a set B which is the largest subset of A that is relatively open in E. b) Prove that every subset A ⊂ E is contained in a set B which is the smallest closed set containing A that is relatively closed in E.
Complete the proof of Theorem 8.36 by verifying (10).
Prove that if E ⊂ R is connected, then E° is also connected. Show that this is false if "R" is replaced by "R2."
Suppose that E ⊂ R" is connected and that E ⊂ A ⊂ . Prove that A is connected.
A set A is called clopen if and only if it is both open and closed.a) Prove that every Euclidean space has at least two clopen sets.b) Prove that a proper subset E of Rn is connected if and only if it contains exactly two relatively clopen sets.c) Prove that every nonempty proper subset of Rn has a
Show that Theorem 8.37 is best possible in the following sense.a) There exist sets A, B in R such that (A U B)° ± A° ‹ƒ B".c) There exist sets A, B in R such that Ï‘(A U B) ‰ Ï‘A U Ï‘B and Ï‘(A ˆ© B) ‰ Ï‘A U Ï‘B.
Using Definition 9.1i, prove that the following limits exist,a)b) c)
Using limit theorems, find the limit of each of the following vector sequences.a)b) c)
Suppose that xk → 0 in Rn as k → ∞ and that yk is bounded in Rn.a) Prove that xk ∙ yk → ■ 0 as k ∞.b) If n = 3, prove that x* x y* -> 0 as A: -> oo.
Suppose that a ∈ Rn, that xk → a, and that xk - yk → 0, as k → ∞. Prove that yk → a as k → ∞.
a) Prove Theorem 9.4i and ii. b) Prove Theorem 9.4iii and iv. c) Prove Theorem 9.4v. d) Prove Theorem 9.6.
Let E be a nonempty subset of Rn.a) Show that a sequence xk ∈ E converges to some point a ∈ E if and only if for every set U, which is relatively open in E and contains a, there is an N ∈ N such that xk; ∈ U for k > N.b) Prove that a set C ⊂ E is relatively closed in E if and only if
a) A subset E of Rn is said to be sequentially compact if and only if every sequence xk ∈ E has a convergent subsequence whose limit belongs to E. Prove that every closed ball in Rn is sequentially compact.b) Prove that Rn is not sequentially compact.
a) Let E be a subset of Rn. A point a ∈ Rn is called a cluster point of E if E ∩ Br (a) contains infinitely many points for every r > 0. Prove that a is a cluster point of E if and only if for each r > 0, E ∩ Br(a)\{a} is nonempty.b) Prove that every bounded infinite subset of Rn has at
Suppose that E is a bounded noncompact subset of Rn and that f: E †’ (0, ˆž). If there is a g: †’ E †’ R such that g(x) > f(x) for all x ˆˆ E, then prove that there exist x1,..., xN ˆˆ E such that
Suppose that £ is a compact subset of R. If for every x ∈ E there exist a nonnegative function f = fx and an r = r(x) > 0 such that f is C∞ on R, f(t) = 1 for t ∈ (x - r, x + r), and f(t) = 0 for t ∉ (x - 2r, x + 2r), prove that there exist a differentiable function f, a nonzero constant
Suppose that K is compact in Rn and that for every x ∈ K there is an r = r(x) > 0 such that Br(x) ∩ K = {x}. Prove that K is a finite set.
Let E be closed and bounded in R, and suppose that for each x ∈ E there is a function fx, nonnegative, nonconstant, increasing, and C∞ on R, such that fx(x) > 0 and fʹx(y) = 0 for y ∉ E. Prove that there exists a nonnegative, nonconstant, increasing C∞ function f on R such that f(y) > 0
Suppose that f: Rn → Rn and that a ∈ K, where K is a compact, connected subset of Rn. Suppose further that for each x ∈ K there is a δx > 0 such that f(x) = f(y) for all y ∈ Bδx(x). Prove that f is constant on K; that is, if a ∈ K, then f(x) = f(a) for all x ∈ K.
Define the distance between two nonempty subsets A and B of R" bya) Prove that if A and B are compact sets which satisfy A ˆ© B = Î¸, then dist(A, B) > 0.b) Show that there exist nonempty, closed sets A, B in R2 such that A ˆ© B = Î¸ but dist(A, B) = 0.
Suppose that E and V are subsets of R with E bounded, V open, and ⊂ V. Prove that there is a C∞ function F: E → R such that F(x) > 0 for x ∈ E and f(x) = 0 for ∉ V.
For each of the following functions, find the maximal domain of f, prove that the limit of f exists as (x, y) †’ (a, b), and find the value of that limit. (You can prove that the limit exists without using Îµ's and Î´s - see Example 9.17.)a)b) c) d)
Compute the iterated limits at (0, 0) of each of the following functions. Determine which of these functions has a limit as (x, y) (0, 0) in R2, and prove that the limit exists.a)b) c)
Prove that each of the following functions has a limit as (x, y) †’ (0, 0).a)b) where a is ANY positive number.
A polynomial on Rn of degree N is a function of the formwhere aj1 ˆ™ ˆ™ ˆ™ jn are scalars, N1,..., Nn are nonnegative integers, and N = N1 + N2 + ˆ™ ˆ™ ˆ™ + Nn. Prove that if P is a polynomial on Rn and a ˆˆ Rn, then
Suppose that a ∈ Rn that L ∈ Rm, and that f: Rn → Rm. Prove that if f(x) → L as x → a, then there is an open set V containing a and a constant M > 0 such that ||f(x)|| < M for all x ∈ V.
Suppose that a = (a1 ∙ ∙ ∙ an) ∈ Rn, that fj: R → R for j = 1, 2,..., n, and that g(x1, x2,..., xn) := f1(x1) ∙ ∙ ∙ fn(xn).a) Prove that if fj(t) fj(a) as t → aj, for each j = 1,..., n, then g(x) → f1(a1)∙ ∙ ∙ fn(an) as x → a.b) Show that the limit of g might not exist
Suppose that g : R → R is differentiable and that g'(x) > 1 for all x ∈ R. Prove that if g(l) = 0 and f(x, y) = (x - l)2(y + l)/(yg(x)), then there is an L ∈ R such that f(x, y) → L as (x, y) (1, b) for all b ∈ R\{0}.
Define f and g on R by f(x) = sin x and g(x) = x/|x| if x ≠ 0 and g(0) = 0.a) Find f(E) and g(E) for E = (0, π), E = [0, π], E = (-1, 1), and E = [-1, 1]. Compare your answers with what Theorems 9.26, 9.29, and 9.30 predict. Explain any differences you notice.b) Find f-1(E) and g-1(E) for E =
a) A set E ⊂ Rn is said to be polygonally connected if and only if any two points a, b ∈ E can be connected by a polygonal path in E; that is, there exist points xk ∈ E, k = 1,. . . . . N, such that x0 = a, xN = b and L(xk-1; xk) ⊂ E for k = 1,..., N. Prove that every polygonally connected
Define f on [0, ∞) and g on R by f(x) = √x and g(x) = 1/x if x ≠ 0 and g(0) = 0.a) Find f(E) and g(E) for E = (0, 1), E = [0, 1), and E = [0, 1]. Compare your answers with what Theorems 9.26, 9.29, and 9.30 predict. Explain any differences you notice.b) Find f-l(E) and g-1(E) for E = (-1, 1)
Suppose that A is open in Rn and f: A → Rm. Prove that f is continuous on A if and only if f-l (V) is open in Rn for every open subset V of Rm.
Suppose that A is closed in Rn and f: A → Rm. Prove that f is continuous on A if and only if f-l (E) is closed in Rn for every closed subset E of Rm.
Suppose that E ⊂ Rn and that f: E → Rm. a) Prove that f is continuous on E if and only if f-1(B) is relatively closed in E for every closed subset B of Rm. b) Suppose that f is continuous on E. Prove that if V is relatively open in f(E), then f-l (V) is relatively open in E, and if B is
Prove thatis continuous on R2.
Let H be a nonempty, closed, bounded subset of Rn.a) Suppose that f : H †’ Rm is continuous. Prove thatis finite and there exists an x0 ˆˆ H such that ||f(x0)|| = ||f||H. b) A sequence of functions fk: H †’ Rm is said to converge uniformly on H to a function f: H
Let E ⊂ Rn and suppose that D is dense in E (i.e., that D ⊂ E and = E). If f: D → Rm is uniformly continuous on D, prove that f has a continuous extension to E; that is, prove that there is a continuous function g : E → Rm such that g(x) = f(x) for all x ∈ D.
Let E be a connected subset of Rn. If f : E → R is continuous, f(a) ≠ f(b) for some a, b ∈ E, and y is a number which lies between f(a) and f(b), then prove that there is an x ∈ E such that f(x) = y. (You may use Theorem 8.30.)
Identify which of the following sets are compact and which are not. If E is not compact, find the smallest compact set H (if there is one) such that E ⊂ H.a) [1/k : k ∈ N} U {0}b) {(x, y) ∈ R2 : a < x2 + y2 < b) for real numbers 0 < a < bc) {(x, y) ∈ R2 : y = sin(l/x) for some x
Let A, B be compact subsets of Rn. Using only Definition 9.10ii, prove that AU B and A ∩ B are compact.
Suppose that E ⊂ R is compact and nonempty. Prove that sup E, inf E ∈ E.
Suppose that {Va}a∈A is a collection of nonempty open sets in Rn which satisfies Va ∩ Vβ = θ for all α ≠ β in A. Prove that A is countable. What happens to this result when open is omitted?
Prove that if V is open in Rn, then there are open balls B1, B2,..., such thatProve that every open set in R is a countable union of open intervals.
Let n ∈ N.a) A subset E of Rn is said to be sequentially compact if and only if every sequence in E has a convergent subsequence. whose limit belongs to E. Prove that every compact set is sequentially compact.b) Prove that every sequentially compact set is closed and bounded.c) Prove that a set E
Let H ⊂ Rn. Prove that H is compact if and only if every cover {Ea}a∈A of H, where the Eα's are relatively open in H, has a finite subcovering.
Suppose that f, fk: R → R are continuous and nonnegative. Prove that if f(x) → 0 as x → ±∞ and fk ↑ f everywhere on R, then fk → f uniformly on R.
For each of the following functions, find a formula for Ï‰f(t).a)b) c)
Prove that (1 - x/k)k → e-x uniformly on any closed, bounded subset of R.
Show that if f: [a, b] → R is integrable and g: f([a, b]) → R is continuous, then g o f is integrable on [a, b]. (Notice by Remark 3.34 that this result is false if g is allowed even one point of discontinuity.)
Using Theorem 7.10 or Theorem 9.30, prove that each of the following limits exists. Find a value for the limit in each case.a)b) where f is continuously differentiable on [0, 1] and f'(0) > 0. c) d)
a) Prove that for every Îµ > 0 there is a sequence of open intervals {Ik}kˆˆN which covers [0, 1] ˆ© Q such thatb) Prove that if {Ik}kˆˆN is a sequence of open intervals which covers [0, 1], then there is an N ˆˆ N such that
Let E1 be the unit interval [0, 1] with its middle third (1/3, 2/3) removed (i.e., E1 = [0, l/3]U[2/3, 1]). Let E2 be E1 with its middle thirds removed; that is,E2 = [0, 1/9] U [2/9, 1/31 U [2/3, 7/9] U [8/9, 1].Continuing in this manner, generate nested sets Ek such that each Ek is the union of 2k
a) A subset E of X is said to be sequentially compact if and only if every sequence xn ∈ E has a convergent subsequence whose limit belongs to E. Prove that every sequentially compact set is closed and bounded.b) Prove that R is closed but not sequentially compact.c) Prove that every closed
Prove that {xk} is bounded in X if and only if supk∈N p(xk, a) < ∞ for all a ∈ X.
Let Rn be endowed with the usual metric and suppose that {xk} is a sequence in R" with components xk(j); that is,a) Use Remark 8.7 to prove that {xk} is bounded in Rn if and only if there is a C > 0 such that |xk(j)| b) Let a ˆˆ Rn. Prove that xk: †’ a as n †’ ˆž if and only if xk(j)
a) Let a ∈ X. Prove that if xn = a for every n ∈ N, then xn converges. What does it converge to?b) Let X = R with the discrete metric. Prove that xn → a as n → ∞ if and only if xn = a for large
a) Let [xn) and {yn} be sequences in X which converge to the same point. Prove that p(xn yn) → 0 as n → ∞. b) Show that the converse of part a) is false.
Let (xn) be Cauchy in X. Prove that {xn} converges if and only if at least one of its subsequences converges.
Prove that the discrete space R is complete.
a) Prove that the metric space C[a, b] in Example 10.6 is complete. b) Let ||f||1: = ∫ba |f(x)|dx and define dist(f, g) := ||f - g||1 for each pair f, g ∈ C[a, b]. Prove that this distance function also makes C[a, b] a metric space. c) Prove that the metric space C[a, b] defined in part b) is
a) Show that if x ∈ Br(a), then there is an ∈ > 0 such that the closed ball centered at x of radius ε is a subset of Br{a).b) If a ≠ b are distinct points in X, prove that there is an r > 0 such that Br(a) ∩ Br(b) = θ.c) Show that given two balls Br(a) and Bs(b), and a point x ∈
Find all cluster points of each of the following sets.a) E = R\Qb) E = [a, b), a, b ∈ R, a < bc) E = {(-1)nn : n ∈ N}d) E = {xn : n ∈ N}, where xn → x as n → ∞e) E = {l, 1, 2, 1, 2, 3, 1, 2, 3, 4,...}

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