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

a) A point a in a metric space X is said to be isolated if and only if there is an r > 0 so small that Br(a) = {a}. Show that a point a ∈ X is not a cluster point of X if and only if a is isolated.b) Prove that the discrete space has no cluster points.
Prove that a is a cluster point for some E ⊂ X if and only if there is a sequence xn ∈ E \ {a} such that xn → a as n → ∞.
a) Let E be a nonempty subset of X. 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 R has at least one cluster point.
Prove that if fn ∈ C[a,b], then fn → f uniformly on [a, b] if and only if fn → f in the metric of C[a, b] (see Example 10.6).
Find the interior, closure, and boundary of each of the following subsets of R.a) E = {1/n : n ˆˆ N}b)c) E = ‹ƒ(-n, n)d) E = Q
Let V be a subset of X.a) Prove that V is open in X if and only if there is a collection of open balls {Ba: a ˆˆ A} such thatb) What happens to this result if open is replaced by closed?
Let E ⊂ X be closed.a) Prove that ϑE ⊂ £.b) Prove that ϑE = E if and only if Eo = 0.c) Show that b) is false if is not closed.
Identify which of the following sets are open, which are closed, and which are neither. Find E°, , and ϑE and sketch E in each case.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 <
Let a ∈ X, s < r, V = {x ∈ X : s < p(x, a) < r), and E = {x ∈ X : s < p(x, a) < r). Prove that V is open and E is closed. _
Show that if E is nonempty and closed in X and a ∉ E, then infx∈E p(x, a) > 0.
Show that Theorem 10.40 is best possible in the following sense.a) There exist sets A, B in R such that (A U B)° ‰ A° U 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) ‰ (A ˆ© Ï‘B) U (B ˆ© Ï‘A) U (Ï‘A ˆ© Ï‘B).
This exercise is used many times from Section 10.5 onward. Let Y be a subspace of X. a) Show that a set V is open in Y if and only if there is an open set U in X such that V = U C\Y. b) Show that a set £ is closed in Y if and only if there is a closed set A in X such that £ = A n K.
Let f : R → R. Prove that f is continuous on R if and only if f-1 (I) is open in R for every open interval I.
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 X be a metric space.a) Prove that if E ⊂ X is compact, then E is sequentially compact (see Exercise 10.1.10).b) Prove that if X is separable and satisfies the Bolzano-Weierstrass Property, then a set E ⊂ X is sequentially compact if and only if it is compact.
Let A, B be compact subsets of X. Prove that A U 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 X which satisfies Va ∩ Vβ = θ for all a ≠ β in A. Prove that if X is separable, then A is countable. What happens to this result when open is omitted?
Prove that if V is open in a separable metric space X, then there are open balls B1, B2,... such thatProve that every open set in R is a countable union of open intervals.
Suppose that X is a separable metric space which satisfies the Bolzano-Weierstrass Property, that Y is a complete metric space, and that E is a bounded subset of X. Prove that a function f: E → Y is uniformly continuous on E if and only if f can be continuously extended to E; that is, if and only
Suppose that X satisfies the Bolzano-Weierstrass Property and that A and B are compact subsets of X. Prove that if A ∩ B = θ and ifdist (A, B) := inf{p(x, y) : x ∈ A and y ∈ B},then dist (A, B) > 0. Show that even in the space R2, there exist subsets A and B which are closed and satisfy A
a) Prove that Cantor's Intersection Theorem holds for nested compact sets in an arbitrary metric space; that is, if H1, H2, ... is a nested sequence of nonempty compact sets in X, thenb) Prove that (ˆš2, ˆš3) ˆ© Q is closed and bounded but not compact in the metric space Q introduced in
Prove that the Bolzano-Weierstrass Property does not hold for C[a, b] and ||f|| (see Example 10.6). Namely, prove that if fn(x) = xn then ||fn|| is bounded but ||fnk - f|| does not converge for any f ∈ C[0, 1] and any subsequence {nk}.
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 compact or connected. Explain your answers.b) Sketch a graph of setB1(-2, 0) U B1(2, 0) U {(x, 0): -1 < x < 1},and decide
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
Suppose that {Ea}aˆˆA is a collection of connected sets in a metric space X such that ˆ©aˆˆA ‰ Î¸. Prove thatis connected.
a) Sketch a graph of the set{(x,y): x2 + 2y2 0},and decide whether this set is relatively open or relatively closed in the subspace {(x, y) : y > 0}. Do the same for the subspace {(x, y): x2 + 2y2 b) Sketch a graph of set{(x, y): x2 + y2
a) Prove that the intersection of two connected sets in R is connected. Show that this is false if R is replaced by R2.b) Generalize part a) as follows. If {Ea}aˆˆA is an arbitrary collection of connected sets in R, thenis also connected.
Prove that if E ⊂ R is connected, then Eo is also connected. Show that this is false if R is replaced by R2.
Suppose that E ⊂ X is connected and that E ⊂ A ⊂ E. Prove that A is connected.
Suppose that X and Y are metric spaces and that f : X → Y. If X is compact and connected, and if to every x ∈ X there corresponds an open ball Bx such that x ∈ Bx and f(y) = f(x) for all y ∈ Bx, prove that f is constant on X.
Let H ⊂ X. Prove that H is compact if and only if every cover {Ea)a∈A of H, where the Ea's are relatively open in H, has a finite subcover.
A set E in a metric space is called clopen if it is both open and closed.a) Prove that every metric space has at least two clopen sets.b) Prove that a metric space is connected if and only if it contains exactly two clopen sets.
Let X be a metric space. Prove that X is connected if and only if every nonempty proper subset of X has a nonempty boundary.
Let f(x) = sin x and g(x) = x/[x] if x ≠ 0 and g(0) = 0.a) Find f(£) and g(E) for E = (0, π), E = [0, π], E = (-1, 1), and E = [-1, I]. Compare your answers to what Theorems 10.58, 10.61, and 10.62 predict. Explain any departures from the predictions.b) Find f-1(E) and g-1(E) for E = (0, 1), E
Let 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 to what Theorems 10.58, 10.61, and 10.62 predict. Explain any departures from the predictions. b) Find f-1(E) and g-1(E) for E = (-1, 1) and E = [-1, 1].
Suppose that £ c X and that f : E → Y. a) Prove that f is continuous on E if and only if f-l (A) n £ is relatively closed in £ for all closed sets A in Y. b) Suppose that f is continuous on E. Prove that if V is relatively open in f(E), then f-1(V) is relatively open in E, and if A is
Suppose that H is a nonempty compact subset of X and that Y is a Euclidean space.a) If f: H †’ y is continuous, prove thatis finite and there exists an x0 ˆˆ H such that ||f(x0))||Î³ = ||f||H. b) A sequence of functions fk: H †’ Y is said to converge
Suppose that E is a compact subset of X.a) If f, g : E → Rn are uniformly continuous, prove that f + g and f ∙ g are uniformly continuous. Did you need compactness for both results?b) If g: E → R is continuous on E and g(x) ≠ 0 for x ∈ £, prove that 1/g is a bounded function.c) If f, g:
Suppose that E ⊂ X and that f : E → Y. (a) If f is uniformly continuous on E and xn ∈ E is Cauchy in X, prove that f(xn) is Cauchy in Y. (b) Suppose that D is a dense subspace of X (i.e., that D ⊂ X and = X). If Y is complete and f: D → Y is uniformly continuous on D, prove that f has a
Suppose that X is connected. Prove that if there is a nonconstant, continuous function f: X → R, then X has uncountably many points.
a) Prove that given f ∈ C[a, b], then there is a sequence of one-variable polynomials Pn such that Pn → f uniformly on [a, b] as n → ∞.b) Prove that the metric space C[a, b] (see Example 10.6) is separable.
A polynomial on Rn is a function of the formwhere the aj1,. . . . . .. j n 's are scalars and N1,...,Nn ˆˆ N. Prove that if A is compact in Rn and f ˆˆ C(A), then there is a sequence of polynomials Pk on Rn such that Pk †’ f uniformly on A as k: †’
Let R = [a, b] Ã— [c, d] be a rectangle in R2. A function f is said to have separated variables iffor some scalars ck and functions fk ˆˆ C[a, b], gk ˆˆ C[c, d]. Prove that given f ˆˆ C(R) there is a sequence of functions with separated variables, Pn, such
Use Exercise 10.7.1 to prove that if f ˆˆ C[a, b] andFor k = 0, 1,..., then f(x) = 0 for all x ˆˆ [a, b).
Use Exercise 10.7.3 to prove that if f ˆˆ C([a, b] Ã— [c, d]), then
Let T = [0, 2Ï€).a) Prove that the functionÎ¦(x) := (cos x, sin x)is 1-1 from T onto Ï‘B1(0, 0) Š‚ R2.b) Prove thatp(x, y) := ||Î¦ (x) - Î¦(y)||is a metric on T.c) Prove that a function f is continuous on (T, p) if and only if it is continuous and periodic on [0, 2Ï€]; that is, if
Use Exercise 10.7.6 to prove that if f is continuous and periodic on [0. 2Ï€] andfor k = 0,1,..., then f(x) = 0 for all x ˆˆ [a, b].
Compute all mixed second-order partial derivatives of each of the following functions and verify that the mixed partial derivatives are equal. a) f(x, y) = xey. b) f(x, y) = cos(xy). c) f(x, y) = x + y/x2 + 1
Suppose that f: (0, ˆž) †’ R is continuous and bounded and that £{f} exists at some a ˆˆ (0, ˆž). Leta) Prove that for all N ˆˆ N. b) Prove that the integral ˆ«0ˆž e-(s-a)t(É¸)(t)dt converges uniformly on [b,
Using Exercises 11.1.9 and 11.1.10, find the Laplace transforms of each of the functions tet, t sin πt, and t2 cos t.
For each of the following functions, compute fx and determine where it is continuous.a)b)
Suppose that r > 0, that a ∈ Rn, and that f: Br(a) → Rm. If all first-order partial derivatives of f exist on Br(a) and satisfy fxj(x) = 0 for all x ∈ Br(a) and all j = 1, 2,..., n, prove that f has only one value on
Suppose that H = [a, b] Ã— [c, d] is a rectangle, that f: H †’ R is continuous, and that g: [a, b] †’ R is integrable. Prove thatis uniformly continuous on [c, d].
Evaluate each of the following expressions.a)b) c)
Suppose that f is a continuous real function.a) If ˆ«01 f(x) dx = 1, find the exact value ofb) If f is C1 on R and ˆ«Ï€0 f'(x) sin x dx = e, find the exact value ofc) If ˆ«01 f(ˆšx)ex dx = 6, find the exact value of
Evaluate each of the following expressions.a)b)
a) Prove thatconverges uniformly on (-ˆž,ˆž). b) Prove that ˆ«0ˆž e-xydx converges uniformly on [1, ˆž). c) Prove that ˆ«0ˆž ye-xydx exists for each y ˆˆ [0, ˆž) and converges uniformly on any [a, b]
Prove thata)b) c) d) e)
Suppose, for j = 1, 2,... ,n, that fj are real functions continuously differentiable on the interval (-1, 1). Prove that g(x) := f1(x1) ∙ ∙ ∙ fn(xn) is differentiable on the cube (-1, 1) × (-1, 1) × ∙ ∙ ∙ × (-1, 1).
Let V be open in Rn, a ˆˆ V, and f : V †’ Rm.a) Prove that Duf(a) exists for u = ek if and only if fxk (a) exists, in which caseb) Show that if f has directional derivatives at a in all directions u, then the first-order partial derivatives of f exist at a. Use Example 11.11 to
Let r > 0, (a, b) ˆˆ R2, f: Br(a, b) †’ R, and suppose that the first-order partial derivatives fx and fy exist in Br(a, b) and are differentiable at (a, b).a) Set Î”(h) = f(a + h, b + h) - f(a + h, b) - f(a, b + h) + f(a, b) and prove for h sufficiently small thatfor
Suppose that f, g: R †’ Rm are differentiable at a and there is a Î´ > 0 such that g(x) ‰ 0 for all 0
Prove that f(x, y) = √|x, y| is not differentiable at (0, 0).
Prove thatis not differentiable at (0, 0).
Prove thatis differentiable on R2 for all a
Prove that if a > 1/2, thenis differentiable at (0, 0).
Prove thatis continuous on R2 and has first-order partial derivatives everywhere on R2, but f is not differentiable at (0, 0).
Suppose that T ∈ £(Rn; Rm). Prove that T is differentiable everywhere on Rn with DT(a) = T for a ∈ Rn.
Let r > 0, f: Br(0) → R, and suppose that there exists an a > 1 such that |fx)| < ||x|a for all x ∈ βr(0). Prove that f is differentiable at 0. What happens to this result when a = 1?
For each of the following, prove that f and g are differentiable on their domains, and find formulas for D(f + g)(x) and D(f • g)(x). a) f(x, y) = x - y, g(x, y) = x2 + y2 b) f(x, y) = xy, g(x, y) = x sin x - cos y c) f(x, y) = (cos(xy), x log y), g(x, y) = (y, x) d) f'(x, y, z) = (y, x -z), g(x,
The time T it takes for a pendulum to complete one full swing is given bywhere g is the acceleration due to gravity and L is the length of the pendulum. If g can be measured with a maximum error of 1%, how accurately must L be measured (in terms of percentage error) so that the calculated value of
Suppose thatwhere each variable x, y, z is positive and can be measured with a maximum error of p%. Prove that the calculated value of w also has a maximum error of p%.
For each of the following functions, find an equation of the tangent plane to z = f(x, y) at c.a) f(x, y) = x2 + y2, c = (1, -1, 2)b) f(x, y) = x3y - xy3 c=(l, 1, 0)c) f(x, y, z) = xy + sin z, c = (1, 0, π/2, 1)
Find all points on the paraboloid z = x2 + y2 (see Appendix D) where the tangent plane is parallel to the plane x + y + z = 1. Find equations of the corresponding tangent planes. Sketch the graphs of these functions to see that your answer agrees with your intuition.
Let K be the cone, given by z = √x2 + y2.a) Find an equation of each plane tangent to K. which is perpendicular to the plane x + z = 5. b) Find an equation of each plane tangent to K, which is parallel to the plane x - y + z = I.
Suppose that f: Rn †’ R is differentiable at a and that f(a) ‰ 0.a) Show that for ||h|| sufficiently small, f(a + h) ‰ 0.b) Prove that Df(a)(h)/||h|| is bounded for all h ˆˆ Rn{0}.c) If T := -Df(a)/f2(a), show thatfor ||h|| sufficiently small. d) Prove that
Suppose that V is open in Rn, that f, g: V → R3, and that a ∈ V. If f and g are differentiable at a, prove that f × g is differentiable at a and D(f × g)(a)(y) = f(a) × (Dg(a)(y)) - g(a) × (Df(a)(y)) for ally ∈ Rn.
Compute the differential of each of the following functions. a) z = x2 + y2 b) z = sm(xy). c) z = xy/1 + x2 + y2
Let ω; = x2y + z. Use differentials to approximate Aw as (x, y, z) moves from (1, 2, 1) to (1.01, 1.98, 1.03). Compare your approximation with the actual value of Δω.
Let F: R3 → R and f, g, h: R2 → R be C2 functions. If w = F(x, y, z), where x = f(p, q), y = g(p, q), and z = h(p, q), find formulas for Wp, Wq, and wpp.
Suppose that I is a nonempty, open interval and that f: I → Rm is differentiable on I. If f(I) ⊂ ϑBr(0) for some fixed r > 0, prove that f(t) is orthogonal to fʹ(t) for all t ∈ I.
Let V be open in Rn, a ∈ V, f : V → R, and suppose that f is differentiable at a.a) Prove that the directional derivative Duf(a) exists (see Exercise 11.2.10) for each u ∈ Rn such that ||u|| = 1 and Duf(a) = f(a) ∙ u.b) If f(a) ≠ 0 and θ represents the angle between u and f(a), prove
Let r > 0, let a ˆˆ Rn, and suppose that g: Br(a) Rm is differentiable at a.a) If f: Br(g(a)) †’ R is differentiable at g(a), prove that the partial derivatives of h = f o g are given byfor j = 1, 2,...,n. b) If n = m and f: Br(g(a)) Rn is differentiable at g(a), prove
Suppose that k ˆˆ N and that f: Rn †’ R is homogeneous of order k: that is, that f(px) = pkf(x) for all x ˆˆ Rn and all p ˆˆ R. If f is differentiable on Rn, prove thatFor all x = (x1, ...,xn) ˆˆ Rn.
Let f, g: R †’ R be twice differentiable. Prove that u(x, y) := f(xy) satisfiesand v(x, y) := f(x - y) + g(x + y) satisfies the wave equation; that is,
Let f, g: R2 †’ R be differentiable and satisfy the Cauchy-Riemann equations; that is, thathold on R2. u(r, 0) = f(r cos Î¸, r sin Î¸) and v(r, Î¸) = g(r cos Î¸, r sin Î¸), prove that
Let f: R2 †’ R be C2 on R2 and set u(r, Î¸) = f(r cos Î¸, r sin Î¸). If f satisfies the Laplace equation; that is, ifprove for each r ‰ Î¸ that
Leta) Prove that Î¼ satisfies the heat equation (i.e., Î¼xx - Î¼t = 0 for all t > 0 and x ˆˆ R). b) If a > 0, prove that u(x, t) †’ 0, as t †’ 0+, uniformly for x ˆˆ [a, ˆž).
Let u R †’ [0, ˆž) be differentiable. Prove that for each (x, y, z) ‰ (0, 0, 0),satisfies
Suppose that z = F(x, y) is differentiable at (a, b), that Fy(a, b) ‰ 0, and that I is an open interval containing a. Prove that if f: †’ R is differentiable at a, f(a) = b, and F(x, f(x)) = 0 for all x ˆˆ I, then
a) Write out an expression in powers of (x +1) and (y - 1) for f(x, y) = x2 + xy + y2. b) Write Taylor's Formula for f(x, y) = √x + √y, a = (1, 4), and p = 3. c) Write Taylor's Formula for f(x, y) = exy, a = (0, 0), and p = 4.
Suppose that V is open in R2, that (a, b) ˆˆ V, and that f: V †’ R is C3 on V. Prove that
Suppose that V is open in R2, that H = [a, b] × [0, c] ⊂ V, that μ: V → R is C2 on V, and that μ(x0, t0) > 0 for all (x0, t0) ∈ ϑH. a) Show that, given ε > 0, there is a compact set K ⊂ such that u(x, t) > -ε for all (x, t) ∈ H\K. b) Suppose that μ(x1. t1) = - < 0 for some (x1,
a) Prove that every convex set in Rn is connected. b) Show that the converse of part a) is false. c) Suppose that f: R → R. Prove that f is convex (as a function) if and only if E: = {(x, y): y > f(x)} is convex (as a set in R2).
Suppose that f : R2 †’ R is Cp on Î²r(x0, y0) for some r > 0. Prove that, given (x, y) ˆˆ Br(x0, y0), there is a point (c, d) on the line segment between (x0, y0) and (x, y) such that
Suppose that f: Rn †’ R and g: Rn †’ Rn are differentiable on Rn and that there exist r > 0 and a ˆˆ Rn such that Dg(x) is the identity matrix, I, for all x ˆˆ Br(a). Prove that there is a function h: Br(a) {a} †’ Br(x) such thatfor all x
Suppose that V is convex and open in Rn and that f: V → Rn is differentiable on V. If there exists an a ∈ V such that Df(x) = Df(a) for all x ∈ V, prove that there exist a linear function S ∈ £(Rn; Rn) and a vector c ∈ Rn such that f(x) = S(x) + c for all x ∈ V.
Let p ˆˆ N, V be an open set in Rn, x, a ˆˆ V, and f: V †’ R be Cp on V. If L(x; a) Š‚ V and h = x - a, prove that
Let r > 0, a, b ∈ R, f: Br(a, b) → R be differentiable, and (x, y) ∈ Br(a, b). a) Let g(t) = f(tx + (1 - t)a, y) + f{a, ty + (1 - t)b) and compute the derivative of g. b) Prove that there are numbers c between a and x, and d between b and y such that f(x, y) - f(a, b) = {x - a)fx(c, y) + (y -

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