 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 9701 students have viewed full stepbystep solutions from this chapter. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484.

Causation
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable

Complex number
An expression a + bi, where a (the real part) and b (the imaginary part) are real numbers

Conditional probability
The probability of an event A given that an event B has already occurred

Continuous at x = a
lim x:a x a ƒ(x) = ƒ(a)

Division algorithm for polynomials
Given ƒ(x), d(x) ? 0 there are unique polynomials q1x (quotient) and r1x(remainder) ƒ1x2 = d1x2q1x2 + r1x2 with with either r1x2 = 0 or degree of r(x) 6 degree of d1x2

Frequency
Reciprocal of the period of a sinusoid.

Leibniz notation
The notation dy/dx for the derivative of ƒ.

Mapping
A function viewed as a mapping of the elements of the domain onto the elements of the range

Measure of an angle
The number of degrees or radians in an angle

Midpoint (on a number line)
For the line segment with endpoints a and b, a + b2

Octants
The eight regions of space determined by the coordinate planes.

Perihelion
The closest point to the Sun in a planet’s orbit.

Projection of u onto v
The vector projv u = au # vƒvƒb2v

Reference angle
See Reference triangle

Remainder polynomial
See Division algorithm for polynomials.

Standard unit vectors
In the plane i = <1, 0> and j = <0,1>; in space i = <1,0,0>, j = <0,1,0> k = <0,0,1>

Translation
See Horizontal translation, Vertical translation.

Unit vector
Vector of length 1.

Velocity
A vector that specifies the motion of an object in terms of its speed and direction.

Vertex of a parabola
The point of intersection of a parabola and its line of symmetry.