 3.5.1: Givean exampleof a boundedsequencethat is not a Cauchysequence
 3.5.2: Showdirectlyfromthe d$finiti~ that the followingareCauchysequences
 3.5.3: Show directly from the definition that the following are not Cauchy...
 3.5.4: Show directly from the definition that if (xn) and (yn) are Cauchy ...
 3.5.5: If xn := ..;n, show that (xn) satisfies lim IXn+1 xn I= 0, but tha...
 3.5.6: Let p be a given natural number.Give an exampleof a sequence(xn) th...
 3.5.7: Let (xn) be a Cauchy sequence such that xn is an integer for every ...
 3.5.8: Show directly that a bounded, monotone increasing sequence is a Cau...
 3.5.9: If 0 < r < 1 and IXn+1 xnI< rn for all n e N, showthat (xn)is a Ca...
 3.5.10: If xI < x2 are arbitrary real numbers and xn := ~(xn_2 + xn_l) for ...
 3.5.11: If YI < Y2 are arbitrary real numbers and Yn := tYn1 + jYn2 for n...
 3.5.12: If xI > 0 and xn+1 := (2 + xn)I for n 2: 1, show that (xn) is a co...
 3.5.13: If xI := 2 and xn+1:= 2 + l/x~ for n 2: 1, show that (xn) is a cont...
 3.5.14: The polynomial equation x3  5x + 1=0 has a root r with 0 < r < 1. ...
Solutions for Chapter 3.5: The Cauchy Criterion
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 3.5: The Cauchy Criterion
Get Full SolutionsThis textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3. Chapter 3.5: The Cauchy Criterion includes 14 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 14 problems in chapter 3.5: The Cauchy Criterion have been answered, more than 2540 students have viewed full stepbystep solutions from this chapter. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484.

Additive inverse of a real number
The opposite of b , or b

Ambiguous case
The case in which two sides and a nonincluded angle can determine two different triangles

Angle of depression
The acute angle formed by the line of sight (downward) and the horizontal

Center
The central point in a circle, ellipse, hyperbola, or sphere

Confounding variable
A third variable that affects either of two variables being studied, making inferences about causation unreliable

Domain of a function
The set of all input values for a function

Frequency (in statistics)
The number of individuals or observations with a certain characteristic.

Inequality
A statement that compares two quantities using an inequality symbol

Inverse properties
a + 1a2 = 0, a # 1a

kth term of a sequence
The kth expression in the sequence

Linear combination of vectors u and v
An expression au + bv , where a and b are real numbers

Lower bound of f
Any number b for which b < ƒ(x) for all x in the domain of ƒ

Magnitude of an arrow
The magnitude of PQ is the distance between P and Q

Phase shift
See Sinusoid.

Piecewisedefined function
A function whose domain is divided into several parts with a different function rule applied to each part, p. 104.

Real zeros
Zeros of a function that are real numbers.

Response variable
A variable that is affected by an explanatory variable.

Secant
The function y = sec x.

Singular matrix
A square matrix with zero determinant

Wrapping function
The function that associates points on the unit circle with points on the real number line
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