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

Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.a) Suppose that E is a set. If there exists a function f from E onto N, then E is at most countable.b) A dyadic rational is a point x ˆˆ R such that x = n/2m for
Suppose that A and B are sets and that B is uncountable. If there exists a function which takes A onto B, prove that A is uncountable.
Suppose that A is finite and f is 1-1 from A onto B. Prove that is finite.
Let f : A → and g : B → C and define g o f : A → C by (g o f)(x) := g(f(x)). a) Show that if f, g are 1-1 (respectively, onto), then g o f is 1-1 (respectively, onto). b) Prove that if f is 1-1 from A into B and B0: = {y : y = f(x) for some x ∈ A}, then f-l is 1-1 from B0 onto A. c) Suppose
Suppose that n ∈ N and ϕ : {1. 2,..., n) → {1, 2 «}. a) Prove that ϕ is 1-1 if and only if ϕ is onto. b) [PIGEONHOLE PRINCIPLE] Suppose that E is a finite set and that f : E → E. Prove that f is 1-1 on E if and only if f takes E onto £.
A number x0 ∈ R is called algebraic of degree n if it is the root of a polynomial P(x) = anxn + ............... + a1x + a0, where aj ∈ Z, an ≠ 0, and n is minimal. A number x0 that is not algebraic is called transcendental. a) Prove that if n ∈ N and q ∈ Q, then nq is algebraic. b) Prove
Decide which of the following statements are true and which are false. Prove the true ones and provide a counterexample for the false ones. a) If xn converges, then xn/n also converges. b) If xn does not converge, then xn/n does not converge. c) If xn converges and yn is bounded, then xny
Using the method of Example 2.2i, prove that the following limits exist. a) 2 - 1/n → 2 as n → ∞. b) 1 + π/√n → 1 as n → ∞. c) 3(1 + 1/n) → 3 as n → ∞. d) (2n2 + l)/(3n2) → 2/3 as n → ∞.
Suppose that xn is a sequence of real numbers that converges to 1 as n → ∞. Using Definition 2.1, prove that each of the following limits exists. a) 1 + 2xn → 3 as n → ∞. b) (πxn - 2)/xn → π - 2 as n → ∞. c) (x2n - e)/xn → 1 - e as n → ∞.
For each of the following sequences, find two convergent subsequences that have different limits. a) 3-(-l)n b) (-l)3n + 2 c) (n - (-1)nn - 1/n
Suppose that xn ∊ R. a) Prove that {xn} is bounded if and only if there is a C > 0 such that |xn| < C for all n ∊ N. b) Suppose that {xn} is bounded. Prove that xn/nk → 0, as n → ∞, for all k ∊ N.
Let C be a fixed, positive constant. If {bn} is a sequence of nonnegative numbers that converges to 0, and {xn} is a real sequence that satisfies |xn - a| < Cbn for large n, prove that xn converges to a.
a) Suppose that {xn) and {yn} converge to the same real number. Prove that xn - yn → 0 as n → ∞. b) Prove that the sequence {n} does not converge. c) Show that there exist unbounded sequences xn ≠ yn which satisfy the conclusion of part (a).
Determine which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If xn → ∞ and yn → -∞, then xn + yn → 0 as n → ∞. b) If xn → ∞, then 1/xn → 0 as n → ∞. c) If xn → 0, then 1 /x" →> ∞ as n
Prove that each of the following sequences converges to zero. a) xn = sin(log n + n5 + en2)/n b) xn = 2n/(n2 + π) c) xn = (√2n + l)/(n + √2) d) xn = n/2n
Use Definition 2.14 to prove that each of the following sequences diverges to + ∞ or to - ∞. a) xn = n2 - n b) xn = n - 3n2 c) xn = n2+ 1/n d) xn = n2(2 + sin(n3 + n + 1))
Find the limit (if it exists) of each of the following sequences. a) xn = (2 + 3n - 4n2)/(1 - 2n + 3n2) b) xn = (n3 + n - 2)/(2n3 + n - 2) c) xn = √3n + 2 - √n d) xn = (√4n + 1 - √n - 1)/(√9n + 1 - √n + 2)
a) Prove Theorem 2.12iv. b) Prove Corollary 2.16.
Suppose that x ∊ R, xn > 0, and xn → x as n → ∞. Prove that √xn → √x as n → ∞. [For the case x = 0, use inequality (8) in Section 1.2.]
Prove that given x ∊ R there is a sequence rn ∊ Q such that rn → x as n → ∞.
Suppose that x and y are extended real numbers and that {xn}, {yn}, and {wn} are real sequences. a) [SQUEEZE THEOREM FOR ]. If xn → x and yn → x, as n → ∞, and xn < wn < yn for n ∊ N, prove that wn → x as n → ∞. b) [COMPARISON THEOREM FOR ]. If xn → and yn → y as n → ∞,
Using the result in Exercise 2.2.5, prove the following results. a) Suppose that 0 < x1 < 1 and xn+1 = 1 - √1 - xn for n ∊ N. If xn → x as n → ∞, then x = 0 or 1. b) Suppose that x1 > 3 and xn+1 = 2 + √xn - 2 for n ∊ N. If xn → x as n → ∞, then x = 3. (c) Suppose that x1 > 0 and
a) Suppose that 0 < y < 1/10n for some integer n > 0. Prove that there is an integer 0 < w < 9 such that w/10n+1 < y < w/10n+1 + 1/10n+1.b) Prove that given x ∊ [0, 1) there exist integers 0 < xk < 9 such that for all n ∊ N,c) Prove that given x ∊ [0, 1) there exist
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.a) If xn is strictly decreasing and 0 < xn < 1/2, then xn → 0 as n → ∞.b) Ifthen xn has a convergence subsequence.c) If xn is a strictly increasing
Suppose that x0 ∊ (-1, 0) and xn = √xn-1 + 1 - 1 for n ∊ N. Prove that xn ↑ 0 as n → ∞. What happens when x0 ∊ [- 1, 0]?
Suppose that x0 = 2\/3, y0 = 3,and yn = √xnyn–1 for n e N.a) Prove that xn ↓ x and yn ↑ y, as n → ∞, for some x, y ∊ R.b) Prove that x = y and 3.14155 < x < 3.14161.(The actual value of x is π.)
Suppose that 0 < x1 < 1 and xn+1 = 1 - √1 - xn for n e N. Prove that xn ↓ 0 as n → ∞ and xn+1/xn → 1/2, as n → ∞.
Suppose that x0 > 2 and xn = 2 + √xn-1 - 2 for n ∊ N. Use the Monotone Convergence theorem to prove that either xn → 2 or xn → 3 as n → ∞.
Suppose that x0 ∊ R and xn = (1 + xn-1)/2 for n ∊ N. Use ↑he Monotone Convergence theorem to prove that xn → 1 as n → ∞.
a) Suppose that {xn} is a monotone increasing sequence in R (not necessarily bounded above). Prove that there is an extended real number x such that xn → x as n → ∞. b) State and prove an analogous result for decreasing sequences.
Suppose that E ⊂ R is a nonempty bounded set and that sup E ∉ E. Prove that there exists a strictly increasing sequence {xn} that converges to sup E such that xn ∊ E for all n ∊ N.
Let 0 < y1 < x1 and seta) Prove that 0 < yn < xn for all n e N.b) Prove that yn is increasing and bounded above, and that xn is decreasing and bounded below.c) Prove that 0 < xn+1 – yn+1 < (x1 – y1)/2n for n ∊ N.d) Prove that limn→∞ xn = limn→∞ yn. (This common value
Suppose that x0 = 1, y0 = 0, xn = xn-1 + 2yn-1, and yn = xn-1 + yn-1 for n ∊ N. Prove that x2n - 2y2n = ± 1 for n ∊ N and xn/yn → √2 as n → ∞
Decide which of the following statements are true and which are false. Prove the true ones and provide a counterexample for the false ones. a) If {xn} is Cauchy and {yn} is bounded, then {xnyn} is Cauchy. b) If {xn} and {yn} are Cauchy and yn ≠ 0 for all n ∊ N, then {xn/yn} is Cauchy. c) If
Prove that if {xn} is a sequence that satisfiesfor all w ˆŠ N, then {x"} is Cauchy.
Suppose that xn ∊ Z for n e N. If {x"} is Cauchy, prove that xn is eventually constant; that is, that there exist numbers a ∊ Z and N ∊ N such that xn = a for all n > N.
Suppose that xn and vn are Cauchy sequences in R and that a ∊ R. a) Without using Theorem 2.29, prove that axn is Cauchy. b) Without using Theorem 2.29, prove that xn + yn is Cauchy. c) Without using Theorem 2.29, prove that xnyn is Cauchy.
Let {xn} be a sequence of real numbers. Suppose that for each Îµ > 0 there is an N ˆŠ N such that m > n > N implies |ˆ‘mk=n xk|exists and is finite.
Prove that limn→∞ (- l)k/k exists and is finite.
Let {xn} be a sequence. Suppose that there is an a ∊ (0, 1) such that |xn+1 - xn| < an for all n∊N. Prove that xn → x for some x ∊ R.
a) Let E be a subset of R. A point a ∊ R is called a cluster point of E if E ∩ (a - r, a + r) 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 ∩ (a - r, a + r)\{a) is nonempty, b) Prove that every bounded infinite subset of
a) A subset E of R 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 closed bounded interval is sequentially compact. b) Prove that there exist bounded intervals in R that are not sequentially
Find the limit infimum and the limit supremum of each of the following sequences. a) xn = 3-(-1)n b) xn = cos {nπ/2) c) xn = (-l)n+1 + (-!)n/n d) xn = √1 + n2/(2n - 5) e) xn = yn/n, where {yn} is any bounded sequence f) xn = n(1 + (-1)n) + n-1((-1)n - 1) g) xn = (n3 + n2 - n + l)/(n2 + 2n + 5)
Suppose that {xn} is a real sequence. Prove thatand
Let {xn} be a real sequence and r ˆŠ R.a) Prove thatfor n large,b) Prove thatfor infinitely many n ˆŠ N.
Suppose that {xn} and {yn} are real sequences.a) Prove thatprovided that none of these sums is of the form ˆž - ˆž.b) Show that if limn†’ˆž xn exists, thenandc) Show by examples that each of the inequalities in part (a) can be strict.
Let {xn} and {yn} be real sequences.a) Suppose that xn > 0 and yn > 0 for each n ˆŠ N. Prove thatprovided that the product on the right is not of the form 0 €¢ ˆž. Show by example that this inequality can be strict.b) Suppose that xn provided that none of these products is of the form
Suppose that xn > 0 and yn > 0 for all n ˆŠ N. Prove that if xn †’ x as n †’ ˆž (x may be an extended real number), thenprovided that none of these products is of the form 0 €¢ ˆž.
Prove thatfor any real sequence {xn}.
Suppose that xn > 0 for n ˆŠ N. Under the interpretation 1/0 = ˆž and 1/ˆž = 0, prove that
Let xn ˆŠ R. Prove that xn †’ 0 as n †’ ˆž if and only if
Let a ∊ R and let f and g be real functions defined at all points x in some open interval containing a except possibly at x = a. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples for the false ones. a) For each n e N, the function
Using Definition 3.1, prove that each of the following limits exists.a)b)c)d)
Decide which of the following limits exist and which do not. Prove that your answer is correct. (You can use well-known facts about the values of tan x, cos x, and log x, e.g., that log x †’ -ˆž as x †’ 0+.)a)b)c)
Evaluate the following limits using results from this section. (You may assume that sin x, 1 - cos x, tan x, and 3ˆšx converge to 0 as x †’ 0.)a)b)c)d)e)
Suppose that f is a real function.a) Prove that ifexists, then |f(x)| †’ |L| as x †’ a.b) Show that there is a function such that, as x †’ a, |f(x)| †’ |L| but the limit of f (x) does not exist.
For each real function f, define the positive part of f byand the negative part of / bya) Prove that f+(x) > 0, f-(x) > 0, f(x) = f+(x) - f-(x% and |f(x)| = f+(x) + f-(x) all hold for every x ˆŠ Dom (f).b) Prove that ifexists, then f+{x) †’ L+ and f-(x) †’ L- as x †’ a.
Let f, g be real functions j and for each x ˆŠ Dom (f) ˆ© Dom (g) define(f ‹ g)(x) :=max{f(x),g(x)} and (f ‹€ g)(x) := mm{f (x), g(x)}.a) Prove thatandfor all x ˆŠ Dom (f) ˆ© Dom (g).b) Prove that ifexist, then (f ‹ g)(x) †’ L ‹ M and (f ‹€ g)(x) †’ L ‹€ M as
Suppose that a ∊ R and / is an open interval which contains a. If f : I → R satisfies f(x) → f(a), as x → a, and if there exist numbers M and m such that m < f(a) < M, prove that there exist positive numbers ∊ and δ such that m + ∊ < f(x) < M - ε for all JC'S which satisfy |JC - a\ < 8
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.a) If f(x) †’ ˆž as x †’ ˆž and g(x) > 0, then g(x)/f(x) †’ 0 as x †’ ˆž.b) If f(x) †’ 0 as x †’ a+ and g(x) > l for all x
For each of the following, use definitions (rather than limit theorems) to prove that the limit exists. Identify the limit in each case.a)b)c)d)e)
Assuming that ex †’ ea, sin x †’ sin a, and cos x †’ cos a as x †’ a for any aˆŠR, evaluate the following limits when they exist.a)b)c)d)e)f.
Recall that a polynomial of degree n is a function of the formP{x) =anxn + an-1 + ˆ™ ˆ™ ˆ™ ˆ™ + a1x + a0,where aj ˆŠ R for j = 0,1,..., n and an ‰ 0.a) Prove that if 00 = 1, then limx†’a xn = an for n = 0,1, ˆ™ ˆ™ ˆ™ and a ˆŠ R.b) Prove that if P is a
Prove the following comparison theorems for real functions f and g, and a ˆŠ R.a) If f(JC) > g(x) and g(x) †’ ˆž as x †’ a, then f(x) †’ ˆž as x †’ a.b) If f(x)then g(x) †’ L as †’ ˆž .
Prove the following special case of Theorem 3.17: Suppose that f : [a, ∞) → R for some a ∊ R. Then f(x) → L as x → ∞ if and only if f(xn) → L for any sequence xn ∊ (a, ∞) which converges to ∞ as n → ∞ .
Suppose that f : [0, 1] R and f(a) = limx→a f(x) for all a ∊ [0,1]. Prove that f(q) = 0 for all q ∊ Q ∩ [0, 1] if and only if f(x) = 0 for all x ∊ [0, I].
Suppose that P is a polynomial and that P(a) > 0 for a fixed a ˆŠ R. Prove that P(x)/(x - a) †’ ˆž as x †’ a+, P(x)/(x - a) †’ -ˆž as x †’ a-, butdoes not exist.
Suppose that f: N †’ R. Ifprove that limn†’ˆž f(n)/n exists and equals L.
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If f is continuous on [a, b] and J := f([a, b]), then J is a closed, bounded interval. b) If f and g are continuous on [a, b], if f(a) < g(a) and f(b) >
Use limit theorems to show that the following functions are continuous on [0,1].a)b)c)d)
If f : R †’ R is continuous andprove that f has a minimum on R; that is, there is an xm ˆŠ R such that
Let a > 1. Assume that ap+q = apaq and (ap)q = apq for all p,q ˆŠ Q, and that ap For each x ˆŠ R, definea) Prove that A(x) exists and is finite for all x ˆŠ R, and that A(p) = ap for all p ˆŠ Q. Thus ax:= A(x) extends the "power of a" function from Q to R.b) If x, y ˆŠ R with x c) Use
For each of the following, prove that there is at least one x ∊ R which satisfies the given equation. a) ex = x3 b) ex = 2cos x + 1 c) 2x = 2 - 3x
If f : [a, b] → [a, b] is continuous, then f has a fixed point; that is, there is a c ∊ [a, b] such that f(c) = c.
Show that there exist nowhere-continuous functions f and g whose sum f + g is continuous on R. Show that the same is true for the product of functions.
Suppose that f : R → R satisfies f(x + y) = f(x) + f(y) for each x, y ∊ R. a) Show that f(nx) = nf(x) for all x ∊ R and n ∊ Z. b) Prove that f(qx) = qf(x) for all x e R and q ∊ Q. c) Prove that f is continuous at 0 if and only if f is continuous on R. d) Prove that if f is continuous at
Suppose that f : R → (0, ∞) satisfies f(x + y) = f(x)/(y). Modifying the outline in Exercise 3.3.8, show that if f is continuous at 0, then there is an a ∊ (0, ∞) such that f(x) = ax for all x ∊ R. (You may assume that the function ax is continuous on R.)
Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If f is uniformly continuous on (0, ∞) and g is positive and bounded on (0, ∞), then fg is uniformly continuous on (0, ∞). b) The function x log(1/x)
Using Definition 3.35, prove that each of the following functions is uniformly continuous on (0,1). a) f(x) = x2 +x b) f(x) = x3 - x + 2 c) f(*) = x sin 2x
Prove that each of the following functions is uniformly continuous on (0, 1). (You may use l'Hopital's Rule and assume that sin x and log x are continuous on their domains.)a)b)c) f(x) = x log xd) f(x) = (1 - x2)1/x
Assuming that sin x is continuous on R, find all real a such that xα sin(l/x) is uniformly continuous on the open interval (0, 1).
a) Suppose that f : [0, ∞) → R is continuous and that there is an L ∊ R such that f(x) → L as x → ∞. Prove that / is uniformly continuous on [0, ∞). b) Prove that f(x) = l/(x2 + 1) is uniformly continuous on R.
Suppose that a ∊ R, that £ is a nonempty subset of R, and that f, g : E → R are uniformly continuous on E. a) Prove that f + g and αf are uniformly continuous on E. b) Suppose that f, g are bounded on E. Prove that fg is uniformly continuous on E. c) Show that there exist functions f,g
a) Let I be a bounded interval. Prove that if f : I → R is uniformly continuous on I, then f is bounded on I. b) Prove that a) may be false if I is unbounded or if I is merely continuous.
Suppose that f is continuous on [a, b]. Prove that given ε > 0 there exist points x0 = a < x1 < ∙ ∙ ∙ < xn = b such that if Ek: = {y : f(x) = y for some x ∊ [xk-1, xk]}, then sup Ek - inf £k < ε for k = 1,2,..., n.
Let ⊂ R. A function f : E → R is said to be increasing on E if and only if x1, x2 ∊ E and x1 < x2 imply f(x1) < f(x2). Suppose that f is increasing and bounded on an open, bounded, nonempty interval (a, b). a) Prove that f{a+) and f(b-) both exist and are finite. b) Prove that / is continuous
Prove that a polynomial of degree n is uniformly continuous on R if and only if n = 0 or 1.
Suppose that f,g : [a, b] → R. Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones. a) If f = g2 and / is differentiable on [a, b], then g is differentiable on (a, b). b) If f is differentiable on [a, b], then f
For each of the following real functions, use Definition 4.1 directly to prove that f'(a) exists. a) f(x) = x2 + x, ∊ R b) f{x) = √x, a > 0 c) f(x) = l/x, a ≠ 0
a) Prove that (x")' = nxn-1 for every n eN and every x e R. b) Prove that (xn")' = nxn-1 for every n ∊ -NU{0} and every x ∊ (0, ∞).
Suppose thatShow that fa{x) is continuous at x = 0 when a > 0 and differentiable at x = 0 when a > 1. Graph these functions for a = 1 and a = 2 and give a geometric interpretation of your results.
Let I be an open interval which contains 0 and f : I → R. If there exists an a > 1 such that |f(x)| < |x|a for all x ∊ I, prove that f is differentiable at 0. What happens when a = 1?
a) Find all points (a, b) on the curve C, given by y = x + sin x, so that the tangent lines to C at (a, b) are parallel to the line y = x + 15. b) Find all points (a, b) on the curve C, given by v = 3x2 + 2, so that the tangent lines to C at (a, b) pass through the point (-1, -7).
Define f on R byFind all n ˆŠ N such that f(n) exists on all of R.
Suppose that f : (0, ∞) → R satisfies f(x) - f(y) = f(x/y) for all x, y ∊ (0, ∞) and f(l) = 0. a) Prove that f is continuous on (0, ∞) if and only if f is continuous at l. b) Prove that f is differentiable on (0, ∞) if and only if f is differentiable at 1. c) Prove that if f is
Let I be an open interval, f : I †’ R, and c ˆŠ l. The function f is said to have a local maximum at c if and only if there is a Î´ > 0 such that f(c) > f(x) holds for all |x - c| a) If f has a local maximum at c, prove thatfor u > 0 and t b) If f is differentiable at c and has a
Suppose that I = (-a, a) for some a > 0. A function f : I → R is said to be even if and only if f(-x) = f(x) for all x ∊ I, and said to be odd if and only if f(-x) = - f(x) for all x ∊ I. a) Prove that if f is odd and differentiable on f, then f' is even on I. b) Prove that if f is even and
Suppose that I is an open interval containing a, and that f, g, h : I †’ R. Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.a) If f g, and h are differentiable at a, then (fgh)Ê¹(a) = f'{a)g{a)h{a) +
Suppose that f and g are differentiable at 2 and 3 with f'{2) = a, f'(3) = ft, g'(2) = c, and g'(3) = d. ft. If f(2) = 1, f(3) = 2, g(2) = 3, and g(3) = 4, evaluate each of the following derivatives. a) (fg)'(2) b) (f/g)'(3) c) (g o f)'(3) d) (fog)'(2)
Suppose that f is differentiable at 2 and 4 with f(2) = 2, f(4) = 3. f'(2) = π, and f'(4) = e. a) If g(x) = xf(x2), find the value of g'(2). b) If g(x) = f2(√x), find the value of g'(4). c) If g(x) = x/f(x3), find the value of gʹ(3√/2).

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