 2.1.1: If b 11]) th & 11 . .~ f  , . a, E~, prove e 10 owmg. riM Gj ~ ...
 2.1.2: Provethalli~,~~'k:ilieg L  ,  6 'L  ~'.. ,.. .. ~ fr(j'tVI. J...
 2.1.3: Solve the following equations, justifying each step by referring to...
 2.1.4: If a E R satisfies a .a = a, provethateithera = 0 ora =
 2.1.5: If a i=0 and b i=0, show that 1/(ab) = (l/a)(l/b). ~ Usethe argume...
 2.1.6: Modify the proof of Theorem 2.1.4 to show that there does not exist...
 2.1.7: (a) Show that if x, y are rational numbers, then x + y and xy are r...
 2.1.8: Let K := {s + t..li : s, t E Q}. Show that K satisfies the followi...
 2.1.9: (a) If a < band e :::d, provethata + e < b + d. (b) If 0 < a < band...
 2.1.10: (a) Show that if a > 0, then l/a > 0 and I/O/a) = a. (b) Show that ...
 2.1.11: Leta, b, e, dbenumbers satisfying 0 < a < bande < d.< O.Give an exa...
 2.1.12: If a, b E JR.,show that a2 + b2 = 0 if and only if a = 0 and b = O. 1
 2.1.13: If 0 ::: a < b, show that a2 ::: ab < b
 2.1.14: Show by example that it does not follow that a2 < ab < b
 2.1.15: If 0 < a < b, show that (a) a < .;ab < b, and (b) l/b < l/a. 1
 2.1.16: Find all real numbers x that satisfy the following inequalities. (a...
 2.1.17: Prove the followingform of Theorem2.1.9:If a E JR.is such that 0 ::...
 2.1.18: Let a, b E JR., and supposethat for everyE> 0 we havea :::b + E.Sho...
 2.1.19: Prove that [~(a + b)]2 ::: ~ (a2 + b2) for all a, bE IR.Show that e...
 2.1.20: (a) If 0 < e < 1, show that 0 < e2 < e < (b) If 1 < e, show that 1 ...
 2.1.21: (a) Prove there is no n EN such that 0 < n < (Use the WellOrdering...
 2.1.22: (a) If e > 1, show that en 2: e for all n EN, and that en > e for n...
 2.1.23: If a > 0, b > 0 and n E N, show that a < b if and only if an < bn. ...
 2.1.24: (a) If e > 1 and m, n E N, show that em > en if and only if m > n. ...
 2.1.25: Assuming the existence of roots, show that if e > 1, then e1/m < e1...
 2.1.26: Use Mathematical Induction to show that if a E JR.and m, n, EN, the...
Solutions for Chapter 2.1: The Algebraic and Order Properties of IR
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 2.1: The Algebraic and Order Properties of IR
Get Full SolutionsSince 26 problems in chapter 2.1: The Algebraic and Order Properties of IR have been answered, more than 3115 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3. Chapter 2.1: The Algebraic and Order Properties of IR includes 26 full stepbystep solutions.

Amplitude
See Sinusoid.

Bar chart
A rectangular graphical display of categorical data.

Branches
The two separate curves that make up a hyperbola

Categorical variable
In statistics, a nonnumerical variable such as gender or hair color. Numerical variables like zip codes, in which the numbers have no quantitative significance, are also considered to be categorical.

Composition of functions
(f ? g) (x) = f (g(x))

Conic section (or conic)
A curve obtained by intersecting a doublenapped right circular cone with a plane

Degree of a polynomial (function)
The largest exponent on the variable in any of the terms of the polynomial (function)

Direction vector for a line
A vector in the direction of a line in threedimensional space

Equal complex numbers
Complex numbers whose real parts are equal and whose imaginary parts are equal.

Focus, foci
See Ellipse, Hyperbola, Parabola.

Horizontal shrink or stretch
See Shrink, stretch.

Index
See Radical.

Initial value of a function
ƒ 0.

Multiplicative inverse of a complex number
The reciprocal of a + bi, or 1 a + bi = a a2 + b2 ba2 + b2 i

nth root
See Principal nth root

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Power regression
A procedure for fitting a curve y = a . x b to a set of data.

Pythagorean identities
sin2 u + cos2 u = 1, 1 + tan2 u = sec2 u, and 1 + cot2 u = csc2 u

Secant
The function y = sec x.

Terminal point
See Arrow.
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