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

Use the method of Example 3.4.3(b) to show that if 0 < c < 1, then lim(c1/n) = 1.
Let (fn) be the Fibonacci sequence of Example 3.1.2(d), and let xn := fn+1/fn. Given that lim(xn) = L exists, determine the value of L.
Let xn := n1/n for n ∈ N. (a) Show that xn+1 < xn if and only if (1 + 1/n)n < n, and infer that the inequality is valid for n > 3. (See Example 3.3.6.) Conclude that (xn) is ultimately decreasing and that x := lim(xn) exists. (b) Use the fact that the subsequence (x2n) also converges to x to
Establish the convergence and find the limits of the following sequences: (a) ((1 + 1/n2)n2), (b) ((1 + 1/2n)n, (c) ((1 + 1/n2)2n2) (d) (1 + 2/n)n.
If x1 < x2 are arbitrary real numbers and xn : = 1/2 (xn-2 + xn-1) for n > 2, show that (xn) is convergent. What is its limit?
Show directly from the definition that the following are not Cauchy sequences. (a) ((-1)n, (b) n + (-1)n/n), (c) (ln n)
Let p be a given natural number. Give an example of a sequence (xn) that is not a Cauchy sequence, but that satisfies lim|xn+p - xn| = 0.
Show directly that a bounded, monotone increasing sequence is a Cauchy sequence.
Establish the proper divergence of the following sequences. (a) (|√n), (b) (√n + 1, (c) (√n - 1), (d) (n/√n + 1.
Let (xn) be properly divergent and let (yn) be such that lim(xnyn) belongs to R. Show that (yn) converges to 0.
Let (xn) and (yn) be sequences of positive numbers such that lim(xn/yn) = 0. (a) Show that if lim(xn) = +∞, then lim(yn) = + ∞. (b) Show that if (yn) is bounded, then lim(xn) = 0.
Investigate the convergence or the divergence of the following sequences: (a) (√n2 + 2), (b) (√n/(n2 + 1), (c) (√n2 + 1/√n, (d) (sin√n.
Let (xn) and (yn) be sequences of positive numbers such that lim(xn/yn) = + ∞, (a) Show that if lim(yn) = + ∞, then lim(xn) = + ∞. (b) Show that if (xn) is bounded, then lim(yn) = 0.
Use the Cauchy Condensation Test to discuss the p-series
Use the Cauchy Condensation Test to establish the divergence of the series: (a) ∑1/n ln n, (b) ∑1/n(ln n)(ln ln n)c,
Show that if c > 1, then the following series are convergent: (a) ∑ 1/n(ln n)c, (b) ∑ 1/n(ln n)(ln ln n)c.
By using partial fractions, show that(a)(b) (c) $By using partial fractions, show that (a) (b) (c)$ $By using partial fractions, show that (a) (b) (c)$ $By using partial fractions, show that (a) (b) (c)$
Let r1, r2, . . . , rn, . . . be an enumeration of the rational numbers in the interval [0,1]. (See Section 1.3.) For a given e > 0, put an interval of length en about the nth rational number rn for n = 1, 2, 3, . . . , and find the total sum of the lengths of all the intervals. Evaluate this
The sequence (xn) is defined by the following formulas for the nth term. Write the first five terms in each case: (a) xn := 1 + (∙1)n, (b) xn := (-1)n/n; (c) xn := 1/n(n + 1), (d) x := 1/n2 + 2.
Let b ∈ R satisfy 0 < b < 1. Show that lim(nbn) = 0. [Use the Binomial Theorem as in Example 3.1.11(d).]
Show that lim(2n/n!) = 0. [If n > 3, then 0 < 2n/n! < 2(2/3)n-2.]
The first few terms of a sequence (xn) are given below. Assuming that the ''natural pattern'' indicated by these terms persists, give a formula for the nth term xn. (a) 5, 7, 9, 11, . . . , (b) 1/2, -1/4, 1/8, -1/16, . . . , (c) 1/2, 2/3, 3/4, 4/5, . . . , (d) 1, 4, 9, 16, . . . .
List the first five terms of the following inductively defined sequences. (a) x1 := 1; xn+1 := 3xn + 1; (b) y1 := 2; yn+1 := 1/2 (yn + 2/yn); (c) z1 := 1; z2 := 2; zn+2 := (zn+1 + zn)/(zn+1 - zn); (d) s1 := 3; s2 := 5; sn+2 := sn + sn+1:
Use the definition of the limit of a sequence to establish the following limits. (a) lim(n/n2 + 1) = 0; (b) lim(2n/n + 1) = 2; (c) lim(3n + 1/2n + 5) = 3/2; (d) lim(n2 - 1/2n2 + 3) = ½.
6. Show that (a) lim(1/√n + 7) = 0; (b) lim(√n/n + 2) = 2; (c) lim(√n/n + 1) = 0; (d) lim(-1)nn/n2 + 1) = 0: 7. Let xn := 1/ln(n + 1) for n ∈ N. (a) Use the definition of limit to show that lim(xn) = 0. (b) Find a specific value of K(ε) as required in the definition of limit for each of
Let xn := 1/ln(n + 1) for n ∈ N. (a) Use the definition of limit to show that lim(xn) = 0. (b) Find a specific value of K(ε) as required in the definition of limit for each of (i) ε = 1/2, and (ii) ε = 1/10.
For xn given by the following formulas, establish either the convergence or the divergence of the sequence X = (xn). (a) xn := n/n + 1; (b) xn := (-1)nn/n + 1; (c) xn := n2/n + 1; (d) xn := 2n2 + 3/n2 + 1.
If a > 0; b > 0; show that lim (√n + a)(n + b) - n) = (a + b)/2.
Apply Theorem 3.2.11 to the following sequences, where a, b satisfy 0 < a < 1; b > 1. (a) (an); (b) (bn/2n); (c) (n/bn); (d) (23n/32n).
Let X = (xn) be a sequence of positive real numbers such that lim(xn+1/xn) = L > 1. Show that X is not a bounded sequence and hence is not convergent.
Discuss the convergence of the following sequences, where a, b satisfy 0 < a < 1; b > 1. (a) (n2an); (b) (bn/n2); (c) (bn/n!); (d) (n!/nn):
Let (xn) be a sequence of positive real numbers such that lim(x1/nn) = L < 1. Show that there exists a number r with 0 < r < 1 such that 0 < xn < rn for all sufficiently large n ∈ N. Use this to show that lim(xn) = 0.
Find the limits of the following sequences: (a) lim((2 + 1/n)2). (b) lim((-1)n/n + 2). (c) lim(√n - 1/√n + 1). (d) lim(n + 1/n√n).
If (bn) is a bounded sequence and lim(an) = 0, show that lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used.
Let yn := √n + 1 - √n for n ∈ N. Show that (√nyn) converges. Find the limit.
Let x1:= 8 and xn+1 := 1/2 xn + 2 for n ∈ N. Show that (xn) is bounded and monotone. Find the limit.
Establish the convergence or the divergence of the sequence (yn), where yn := 1/n + 1 + 1/n + 2 +∙ ∙ ∙ ∙+ 1/2n for n ∈ N:
Let xn := 1/12 + 1/22 +∙ ∙ ∙+1/n2 for each n ∈ N. Prove that (xn) is increasing and bounded, and hence converges. [If k > 2, then 1/k2 1/k(k - 1) = 1/(k - 1) - 1/k.]
Establish the convergence and find the limits of the following sequences. (a) ((1 + 1/n)n+1), (b) ((1 + 1/n)2n), (c) ((1 + 1/n + 1)n), (d) ((1 - 1/n)n.
Let x1 > 1 and xn+1 := 2 - 1/xn for n ∈ N. show that (xn) is bounded and monotone. Find the limit.
Let x1 > 2 and xn+1 := 1 +√xn - 1 for n ∈ N. Show that (xn) is decreasing and bounded below by 2. Find the limit.
Determine a condition on |x - 1| that will assure that: (a) |x2 - 1| < 1/2, (b) |x2 - 1 < 1/10-3, (c) |x2 - 1| < 1/n for a given n ∈ N, (d) |x3 - 1| < 1/n for a given n ∈ N.
Use the definition of limit to show that(a)(b)
Use the definition of limit to prove the following.(a)(b)
Show that the following limits do not exist.(a)(b) (c) (d)
(b) Show by example that if L ‰ 0, then f may not have a limit at c.
Let f : R → R be defined by setting f(x) := x if x is rational, and f(x) = 0 if x is irrational.(a) Show that f has a limit at x = 0.(b) Use a sequential argument to show that if c ≠ 0, then f does not have a limit at c.
Let f : R → R, let I be an open interval in R, and let c ∈ I. If f1 is the restriction of f to I, show that f1 has a limit at c if and only if f has a limit at c, and that the limits are equal.
Determine a condition on |x - 4| that will assure that: (a) |√x - 2| < 1/2, (b) |√x - 2| < 10-2.
Let f := R †’ R and let Š‚ R. Show that
Let I := (0, a) where a > 0, and let g(x) := x2 for x ˆˆ I. For any points x, c ˆˆ I, show that |g(x) - c2|
Show that
Use either the Îµ- Î´ definition of limit or the Sequential Criterion for limits, to establish the following limits.(a)(b) (c) (d)
Give examples of functions f and g such that f and g do not have limits at a point c, but such that both f + g and fg have limits at c.
Let f : R †’ R be such that f(x + y) = f(x) + f(y) for all x, y in R. Assume that exists. Prove that L = 0, and then prove that f has a limit at every point c ˆˆ R. [First note that F(2x) = f(x) + f(x) = 2f(x) for x ˆˆ R. Also note that f(x) = f(x - c) + f(c) for x, c in R.]
Functions f and g are defined on R by f (x) := x + 1 and g (x) := 2 if x ‰ 1 and g(1) := 0.
Determine the following limits and state which theorems are used in each case. (You may wish to use Exercise 15 below.)(a)(b) (c) (d)
Prove that does not exist but that
Use the definition of the limit to prove the first assertion in Theorem 4.2.4(a).
Use the sequential formulation of the limit to prove Theorem 4.2.4(b).
Let n ˆˆ N be such that n > 3. Derive the inequality -x2 to show that
Suppose that where L > 0, and that Show that If L = 0, show by example that this conclusion may fail.
Evaluate the following limits, or show that they do not exist.(a)(b)(c)(d)(e)(f)(g)(h)
Define g : R → R by g(x) := 2x for x rational, and g(x) := x + 3 for x irrational. Find all points at which g is continuous.
Let A := (0,1) and let k : A → R be defined as follows. For x ∈ A, x irrational, we define k(x) = 0; for x ∈ A rational and of the form x = m/n with natural numbers m, n having no common factors except 1, we define k(x) := n. Prove that k is unbounded on every open interval in A. Conclude
Let f : (0; 1) †’ R be bounded but such that does not exist. Show that there are two sequences (xn) and (yn) in (0, 1) with lim(xn) = 0 = lim(yn), but such that (f(xn)) and (f(yn)) exist but are not equal.
Let a < b < c. Suppose that f is continuous on [a, b], that g is continuous on [b, c], and that f (b) = g(b). Define h on [a, c] by h(x) := f(x) for x ∈ |a, b and h(x) := g(x) for x ∈ |b, c. Prove that h is continuous on [a, c].
If x ∈ R, we define [[x]] to be the greatest integer n ∈ Z such that n < x. (Thus, for example, [[8:3]] = 8; [[π]] = 3; [[- π]] = -4.) The function x → [[x]] is called the greatest integer function. Determine the points of continuity of the following functions: (a) f(x) := [[x]], (b) g(x)
Let A ⊂ R and let f : A → R be continuous at a point c ∈ A. Show that for any ε > 0, there exists a neighborhood Vδ(c) of c such that if x; y ∈ A ∩ Vδ(c), then |f(x) - f(y)| < ε.
Let A ⊂ B ⊂ R, let f : B → R and let g be the restriction of f to A (that is, g(x) = f(x) for x ∈ A). (a) If f is continuous at c ∈ A, show that g is continuous at c. (b) Show by example that if g is continuous at c, it need not follow that f is continuous at c.
Determine the points of continuity of the following functions and state which theorems are used in each case. (a) f(x) := x2 + 2x + 1/x2 + 1 (x ∈ R), (b) g(x) := √x + √x (x > 0), (c) h(x) := √1 + |sin x|/x (x ≠ 0), (d) k(x) := cos √1 + x2 (x ∈ R):
A function f : R → R is said to be additive if f(x + y) = f(x) + f(y) for all x, y in R. Prove that if f is continuous at some point x0, then it is continuous at every point of R.
Suppose that f is a continuous additive function on R. If c := f(1), show that we have f(x) = cx for all x ∈ R. [First show that if r is a rational number, then f(r) = cr.]
Let g : R → R satisfy the relation g(x + y) = g(x) g(y) for all x, y in R. Show that if g is continuous at x = 0, then g is continuous at every point of R. Also if we have g(a) = 0 for some a ∈ R, then g(x) = 0 for all x ∈ R.
Let f, g be defined on R and let c ˆˆ R. Suppose that and that g is continuous at b. Show that lim (Compare this result with Theorem 5.2.7 and the preceding exercise.)
Let I := [a, b and let f : I → R be a continuous function such that f(x) > 0 for each x in I. Prove that there exists a number a > 0 such that f(x) > a for all x ∈ I.
Let I := [a; b], let f : I → R be continuous on I, and assume that f(a) < 0; f(b) > 0. Let W := {x ∈ I : f(x) < 0}, and let w := sup W. Prove that f(w) = 0. (This provides an alternative proof of Theorem 5.3.5.)
Let I := [0; π/2] and let f : I → R be defined by f(x) := sup{x2, cos x} for x ∈ I. Show there exists an absolute minimum point x0 ∈ I for f on I. Show that x0 is a solution to the equation cos x = x2.
Suppose that f : R †’ R is continuous on R and that = 0 and Prove that f is bounded on R and attains either a maximum or minimum on R. Give an example to show that both a maximum and a minimum need not be attained.
Examine which open [respectively, closed] intervals are mapped by f(x) := x2 for x ∈ R onto open [respectively, closed] intervals.
Examine the mapping of open [respectively, closed] intervals under the functions g(x) := 1/(x2 + 1) and h(x) := x3 for x ∈ R.
Let I := [a, b] and let f : I → R be a (not necessarily continuous) function with the property that for every x ∈ I, the function f is bounded on a neighborhood Vδx (x) of x (in the sense of Definition 4.2.1). Verify that f is bounded on I.
Let I := [a, b] and let f : I → R be a (not necessarily continuous) function with the property that for every x ∈ I, the function f is bounded on a neighborhood Vδx (x) of x (in the sense of Definition 4.2.1). Prove that f is bounded on I.
Let I := [a, b] and let f : I → R and g : I → R be continuous functions on I. Show that the set E := {x ∈ I : f(x) = g(x)} has the property that if (xn) ⊂ E and xn → x0, then x0 ∈ E.
Let I := [a, b] and let f : I → R be a continuous function on I such that for each x in I there exists y in I such that | f (y)| < ½| f(x)|. Prove there exists a point c in I such that f(c) = 0.
Show that every polynomial of odd degree with real coefficients has at least one real root.
Let f be continuous on the interval [0, 1] to R and such that f(0) = f(1). Prove that there exists a point c in [0, 1/2] such that f(c + 1/2). [Consider g(x) = f(x) - f(x + 1/2).] Conclude that there are, at any time, antipodal points on the earth's equator that have the same temperature.
Prove that if f is uniformly continuous on a bounded subset A of R, then f is bounded on A.
If g(x) := √x for x ∈ (0, 1), show that there does not exist a constant K such that |g(x)| < K|x| for all x ∈ (0, 1). Conclude that the uniformly continuous g is not a Lipschitz function on [0, 1].
Show that if f is continuous on [0, ∞) and uniformly continuous on [a, ∞) for some positive constant a, then f is uniformly continuous on [0, ∞).
A function f : R → R is said to be periodic on R if there exists a number p > 0 such that f(x + p) = f(x) for all x ∈ R. Prove that a continuous periodic function on R is bounded and uniformly continuous on R.
Let f and g be Lipschitz functions on A. (a) Show that the sum f + g is also a Lipschitz function on A. (b) Show that if f and g are bounded on A, then the product fg is a Lipschitz function on A. (c) Give an example of a Lipschitz function f on [0, ∞) such that its square f2 is not a Lipschitz
A function is called absolutely continuous on an interval I if for any ε > 0 there exists a δ > 0 such that for any pair-wise disjoint subintervals (xk, yk) = k = 1, 2, . . . , n, of I such that ∑(xk - yk) < δ we have ∑|f(xk) - f(yk)| < e. Show that if f satisfies a Lipschitz condition on I,
Show that the function f(x):= 1/x2 is uniformly continuous on A: = (1, ∞), but that it is not uniformly continuous on B: = (0, ∞).
Use the Non uniform Continuity Criterion 5.4.2 to show that the following functions are not uniformly continuous on the given sets. (a) f(x) := x2; A := (0; ∞) . (b) g(x) := sin(1/x); B := (0, ∞) .
Show that the function f(x) := 1/(1 + x2) for x ∈ R is uniformly continuous on R.
If f (x) := x and g(x) := sin x, show that both f and g are uniformly continuous on R, but that their product fg is not uniformly continuous on R.
Prove that if f and g are each uniformly continuous on R, then the composite function f o g is uniformly continuous on R.
Let d be the gauge on [0, 1] defined by Î´(0):= 1/4 and Î´(t) := 1/2 t for t ˆˆ (0, 1).(a)(b)
Let I := (a, b) and f : I → R. We say that f is ''locally increasing'' at c ∈ I if there exists δ© > 0 such that f is increasing on I ∩ (c - δ(c), c + δ(c)). Prove that if f is locally increasing at every point of I, then f is increasing on I.

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