 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 2641 students have viewed full stepbystep solutions from this chapter.

Absolute minimum
A value ƒ(c) is an absolute minimum value of ƒ if ƒ(c) ? ƒ(x)for all x in the domain of ƒ.

Average rate of change of ƒ over [a, b]
The number ƒ(b)  ƒ(a) b  a, provided a ? b.

Closed interval
An interval that includes its endpoints

Combination
An arrangement of elements of a set, in which order is not important

Coordinate plane
See Cartesian coordinate system.

Course
See Bearing.

Cube root
nth root, where n = 3 (see Principal nth root),

Frequency
Reciprocal of the period of a sinusoid.

Logistic growth function
A model of population growth: ƒ1x2 = c 1 + a # bx or ƒ1x2 = c1 + aekx, where a, b, c, and k are positive with b < 1. c is the limit to growth

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

Nappe
See Right circular cone.

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.

Product of complex numbers
(a + bi)(c + di) = (ac  bd) + (ad + bc)i

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

Quadric surface
The graph in three dimensions of a seconddegree equation in three variables.

Rigid transformation
A transformation that leaves the basic shape of a graph unchanged.

Sample survey
A process for gathering data from a subset of a population, usually through direct questioning.

Secant line of ƒ
A line joining two points of the graph of ƒ.

Standard form of a polar equation of a conic
r = ke 1 e cos ? or r = ke 1 e sin ? ,

Terms of a sequence
The range elements of a sequence.
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