 8.2.1: Show that the sequence xn /(1 + xn does not converge uniformly on [...
 8.2.2: Prove that the sequence in Example 8.2.1(c) is an example of a sequ...
 8.2.3: Construct a sequence of functions on [0, 1] each of which is discon...
 8.2.4: Suppose Un) is a sequence of continuous functions on an interval I ...
 8.2.5: Let I : R + R be uniformly continuous on R and let In(x) := I(x + ...
 8.2.6: LetIn(x) := 1/(1 + xt forx e [0, 1].FindthepointwiselimitI of the s...
 8.2.7: Suppose the sequence Un) converges uniformly to I on the set A, and...
 8.2.8: Let In(x) := nx/(1 + nx2) for x e A := [0,00), Show that each In is...
 8.2.9: Let In (X) := Xn/n for x E [0,1]. Show that the sequ~nce (In) of di...
 8.2.10: Let gn(x) := enx /n for x ::: 0, n E N. Examine the relation betwe...
 8.2.11: Let I := [a, b] and let Un) be a sequence of functions on I + IRth...
 8.2.12: Show that lim fl2 enx2 dx = O.
 8.2.13: If a > 0, show that limL7r(sinnx)/(nx) dx = O.What happens if a = O?
 8.2.14: Let In(x) := nx/(1 + nx) for x E [0, 1]. Show that Un) converges no...
 8.2.15: Letgn(x):= nx(1 x)n for x E [0,1], n EN. Discuss the convergence o...
 8.2.16: Let {rl, r2, . . . , rn . . .} be an enumeration of the rational nu...
 8.2.17: Let In (x) := 1for x E (0, l/n) and In (x) :=Oelsewherein[O, 1].Sho...
 8.2.18: Let In(x) := xn for x E [0, 1], n EN. Show that Un) is a decreasing...
 8.2.19: Let In(x) := x/n forx E [0,00), n EN. Show that Un) is a decreasing...
 8.2.20: Give an example of a decreasing sequence Un) of continuous function...
Solutions for Chapter 8.2: Interchange of Limits
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 8.2: Interchange of Limits
Get Full SolutionsSince 20 problems in chapter 8.2: Interchange of Limits have been answered, more than 2614 students have viewed full stepbystep solutions from this chapter. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.2: Interchange of Limits includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3.

Circular functions
Trigonometric functions when applied to real numbers are circular functions

Distance (in Cartesian space)
The distance d(P, Q) between and P(x, y, z) and Q(x, y, z) or d(P, Q) ((x )  x 2)2 + (y1  y2)2 + (z 1  z 2)2

Focal width of a parabola
The length of the chord through the focus and perpendicular to the axis.

Infinite sequence
A function whose domain is the set of all natural numbers.

Inverse reflection principle
If the graph of a relation is reflected across the line y = x , the graph of the inverse relation results.

Inverse secant function
The function y = sec1 x

kth term of a sequence
The kth expression in the sequence

Linear combination of vectors u and v
An expression au + bv , where a and b are real numbers

nset
A set of n objects.

Nautical mile
Length of 1 minute of arc along the Earthâ€™s equator.

Order of magnitude (of n)
log n.

Parametric curve
The graph of parametric equations.

Product of matrices A and B
The matrix in which each entry is obtained by multiplying the entries of a row of A by the corresponding entries of a column of B and then adding

Repeated zeros
Zeros of multiplicity ? 2 (see Multiplicity).

Solution of a system in two variables
An ordered pair of real numbers that satisfies all of the equations or inequalities in the system

symmetric about the xaxis
A graph in which (x, y) is on the graph whenever (x, y) is; or a graph in which (r, ?) or (r, ?, ?) is on the graph whenever (r, ?) is

Symmetric matrix
A matrix A = [aij] with the property aij = aji for all i and j

Term of a polynomial (function)
An expression of the form anxn in a polynomial (function).

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

Weights
See Weighted mean.
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