- 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
Additive inverse of a real number
The opposite of b , or -b
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
The central point in a circle, ellipse, hyperbola, or sphere
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.
A statement that compares two quantities using an inequality symbol
a + 1-a2 = 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
A function whose domain is divided into several parts with a different function rule applied to each part, p. 104.
Zeros of a function that are real numbers.
A variable that is affected by an explanatory variable.
The function y = sec x.
A square matrix with zero determinant
The function that associates points on the unit circle with points on the real number line
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