 2.4.1: Show that sup{1  Iln : n eN} =
 2.4.2: If S := {lln  11m:n, meN}, find inf S and supS.
 2.4.3: Let S S;JRbe nonempty.Prove that if a number u in JRhas the propert...
 2.4.4: Let S be a nonemptyboundedset in IR. (a) Let a > 0, andlet as := {a...
 2.4.5: Let X be a nonemptyset and let f: X ~ JRhave boundedrange in IR.If ...
 2.4.6: Let A and B be bounded nonempty subsets of JR,and let A + B := {a +...
 2.4.7: Let X be a nonempty set, and let f and g be defined on X and have b...
 2.4.8: Let X =Y := {x e JR:0 < x < I}. Defineh: X x Y ~ JRby h(x, y) := 2x...
 2.4.9: Perform the computations in (a) and (b) of the preceding exercise f...
 2.4.10: LetXandY be nonemptysetsandleth : X x Y ~ JRhaveboundedrangeinJR.Le...
 2.4.11: LetXand Y be nonemptysetsandleth : X x Y +JRhaveboundedrangeinR L...
 2.4.12: Given any x e JR,show that there exists a unique n e Z such that n ...
 2.4.13: If y > 0, showthat thereexistsn eN suchthat 1/2n < y. 1
 2.4.14: Modifythe argumentin Theorem2.4.7to showthat thereexistsa positiver...
 2.4.15: Modifythe argumentin Theorem2.4.7 to showthat if a > 0, then there ...
 2.4.16: Modify the argument in Theorem 2.4.7 to show that there exists a po...
 2.4.17: Complete the proof of the Density Theorem 2.4.8 by removing the ass...
 2.4.18: If u > 0 is any real number and x < y, show that there exists a rat...
Solutions for Chapter 2.4: Applications of the Supremum Property
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 2.4: Applications of the Supremum Property
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Absolute value of a real number
Denoted by a, represents the number a or the positive number a if a < 0.

Aphelion
The farthest point from the Sun in a planet’s orbit

artesian coordinate system
An association between the points in a plane and ordered pairs of real numbers; or an association between the points in threedimensional space and ordered triples of real numbers

Correlation coefficient
A measure of the strength of the linear relationship between two variables, pp. 146, 162.

Determinant
A number that is associated with a square matrix

Difference of two vectors
<u1, u2>  <v1, v2> = <u1  v1, u2  v2> or <u1, u2, u3>  <v1, v2, v3> = <u1  v1, u2  v2, u3  v3>

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

Expanded form of a series
A series written explicitly as a sum of terms (not in summation notation).

Graph of an equation in x and y
The set of all points in the coordinate plane corresponding to the pairs x, y that are solutions of the equation.

Hypotenuse
Side opposite the right angle in a right triangle.

Inequality symbol or
<,>,<,>.

Mathematical model
A mathematical structure that approximates phenomena for the purpose of studying or predicting their behavior

Mode of a data set
The category or number that occurs most frequently in the set.

Oddeven identity
For a basic trigonometric function f, an identity relating f(x) to f(x).

Open interval
An interval that does not include its endpoints.

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Plane in Cartesian space
The graph of Ax + By + Cz + D = 0, where A, B, and C are not all zero.

Probability of an event in a finite sample space of equally likely outcomes
The number of outcomes in the event divided by the number of outcomes in the sample space.

Union of two sets A and B
The set of all elements that belong to A or B or both.

Upper bound for real zeros
A number d is an upper bound for the set of real zeros of ƒ if ƒ(x) ? 0 whenever x > d.