 3.4.1: aiv<:_~ ~~P1e of an unbounded sequence that has a convergent subseq...
 3.4.2: Use the method of Example 3.4.3(b) to show that if 0 < c < 1, then ...
 3.4.3: Let Un) be the Fibonacci sequence of Example 3.I.2(d), and let xn :...
 3.4.4: Show that the following sequences are divergent. (a) (1  (_1)n + l...
 3.4.5: Let X = (xn) and Y = (Yn) be given sequences, and let the "shuffled...
 3.4.6: Let xn := nl/n for n EN. (a) Show that xn+1< xn if and only if (1 +...
 3.4.7: Establishthe convergenceand find the limits of the followingsequences:
 3.4.8: Determine the limits of the following. (a) (3n) 1/2n), (b) ((1 + 1/...
 3.4.9: Suppose that every subsequence of X = (xn) has a subsequence that c...
 3.4.10: Let (xn)be a boundedsequenceand for each n E N let sn := sup{xk:k 2...
 3.4.11: Supposethatxn 2: 0 for all n EN andthat lim(ltxJ exists.Showthat (...
 3.4.12: Show that if (x ) is unbounded', then there exists a subsequence (x...
 3.4.13: If xn := (_1)n In, find the subsequence of (xn) that is constructed...
 3.4.14: Let (xn) be a bounded sequence and let S := sup{xn:n EN}. Show that...
 3.4.15: Let (I n) be a nested sequence of closed bounded intervals. For eac...
 3.4.16: Give an example to show that Theorem 3.4.9 fails if the hypothesis ...
Solutions for Chapter 3.4: Subsequences and the Bolzano Weierstrass Theorem
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 3.4: Subsequences and the Bolzano Weierstrass Theorem
Get Full SolutionsIntroduction 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. Since 16 problems in chapter 3.4: Subsequences and the Bolzano Weierstrass Theorem have been answered, more than 3126 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.4: Subsequences and the Bolzano Weierstrass Theorem includes 16 full stepbystep solutions.

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.

Common difference
See Arithmetic sequence.

Equivalent arrows
Arrows that have the same magnitude and direction.

First octant
The points (x, y, z) in space with x > 0 y > 0, and z > 0.

Frequency
Reciprocal of the period of a sinusoid.

Function
A relation that associates each value in the domain with exactly one value in the range.

Integrable over [a, b] Lba
ƒ1x2 dx exists.

kth term of a sequence
The kth expression in the sequence

Leaf
The final digit of a number in a stemplot.

Lemniscate
A graph of a polar equation of the form r2 = a2 sin 2u or r 2 = a2 cos 2u.

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

Opposite
See Additive inverse of a real number and Additive inverse of a complex number.

Period
See Periodic function.

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

Response variable
A variable that is affected by an explanatory variable.

Righthand limit of ƒ at x a
The limit of ƒ as x approaches a from the right.

Summation notation
The series a nk=1ak, where n is a natural number ( or ?) is in summation notation and is read "the sum of ak from k = 1 to n(or infinity).” k is the index of summation, and ak is the kth term of the series

Triangular number
A number that is a sum of the arithmetic series 1 + 2 + 3 + ... + n for some natural number n.

Vertex form for a quadratic function
ƒ(x) = a(x  h)2 + k

yzplane
The points (0, y, z) in Cartesian space.
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