 11.1.1: If x e (0, 1), let Exbe as in Example 11.1.3(b).Show that if lu  x...
 11.1.2: Show that the intervals (a, 00) and (00, a) are open sets, and tha...
 11.1.3: Write out the Induction argument in the proof of part (b) of the Op...
 11.1.4: Prove that (0, 1] = n:,(0, 1 + I/n), as asserted in Example 11.1.6(a).
 11.1.5: Showthatthe setNof naturalnumbersis a closedset.
 11.1.6: Show that A = {l/n: n eN} is not a closed set, but that A U (OJis a...
 11.1.7: Show that the set Q of rational numbers is neither open nor dosed
 11.1.8: Show that if G is an open set and F is a closed set, then G\F is an...
 11.1.9: A point x e R is said to be an interior point of A ~ R in case ther...
 11.1.10: A point x e R is said to be a boundary point of A ~ R in case every...
 11.1.11: Show that a set G ~ R is open if and only if it does not contain an...
 11.1.12: Show that a set F ~ R is closed if and only if it contains all of i...
 11.1.13: If A ~ JR,let A0 be the union of all open sets that are contained i...
 11.1.14: Using the notation of the preceding exercise, let A, B be sets in R...
 11.1.15: If A ~ JR,let Abe the intersection of all closed sets containing A...
 11.1.16: Using the notation of the preceding exercise, let A, B be sets in R...
 11.1.17: Give an example of a set A ~ JRsuch that A = 0 and A = R
 11.1.18: Show that if F ~ JRis a closed nonempty set that is bounded above, ...
 11.1.19: If G is open andx E G, showthatthe setsAx and Bx in the proofof The...
 11.1.20: If the set Ax in the proof of Theorem 11.1.9 is bounded below,show ...
 11.1.21: If in the notation used in the proof of Theorem 11.1.9,we have ax <...
 11.1.22: If in the notation used in the proof of Theorem 11).9, we have Ix n...
 11.1.23: Show that each point of the Cantor set IFis a cluster point of IF.
 11.1.24: Show that each point of the Cantor set IFis a cluster point of C(IF).
Solutions for Chapter 11.1: Open and Closed Sets in IR
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 11.1: Open and Closed Sets in IR
Get Full SolutionsChapter 11.1: Open and Closed Sets in IR includes 24 full stepbystep 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. This expansive textbook survival guide covers the following chapters and their solutions. Since 24 problems in chapter 11.1: Open and Closed Sets in IR have been answered, more than 6858 students have viewed full stepbystep solutions from this chapter.

Domain of validity of an identity
The set of values of the variable for which both sides of the identity are defined

Equivalent arrows
Arrows that have the same magnitude and direction.

Extraneous solution
Any solution of the resulting equation that is not a solution of the original equation.

Fundamental
Theorem of Algebra A polynomial function of degree has n complex zeros (counting multiplicity).

Halflife
The amount of time required for half of a radioactive substance to decay.

Imaginary axis
See Complex plane.

Linear correlation
A scatter plot with points clustered along a line. Correlation is positive if the slope is positive and negative if the slope is negative

Numerical derivative of ƒ at a
NDER f(a) = ƒ1a + 0.0012  ƒ1a  0.00120.002

Ordered set
A set is ordered if it is possible to compare any two elements and say that one element is “less than” or “greater than” the other.

Polynomial in x
An expression that can be written in the form an x n + an1x n1 + Á + a1x + a0, where n is a nonnegative integer, the coefficients are real numbers, and an ? 0. The degree of the polynomial is n, the leading coefficient is an, the leading term is anxn, and the constant term is a0. (The number 0 is the zero polynomial)

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

Quadratic formula
The formula x = b 2b2  4ac2a used to solve ax 2 + bx + c = 0.

Radian measure
The measure of an angle in radians, or, for a central angle, the ratio of the length of the intercepted arc tothe radius of the circle.

Reflection through the origin
x, y and (x,y) are reflections of each other through the origin.

Standard form of a polynomial function
ƒ(x) = an x n + an1x n1 + Á + a1x + a0

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.

Variable (in statistics)
A characteristic of individuals that is being identified or measured.

Vertical component
See Component form of a vector.

Xscl
The scale of the tick marks on the xaxis in a viewing window.

zaxis
Usually the third dimension in Cartesian space.