- 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 := tYn-1 + jYn-2 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
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable
An expression a + bi, where a (the real part) and b (the imaginary part) are real numbers
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
Reciprocal of the period of a sinusoid.
The notation dy/dx for the derivative of ƒ.
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
The eight regions of space determined by the coordinate planes.
The closest point to the Sun in a planet’s orbit.
Projection of u onto v
The vector projv u = au # vƒvƒb2v
See Reference triangle
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>
See Horizontal translation, Vertical translation.
Vector of length 1.
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.