Introduction to Real Analysis 3rd Edition solutions

Introduction to Real Analysis 3
Prove the second De Morgan Law [Theorem 1.1.4(b)].

Introduction to Real Analysis 3
If A and B are sets, show that A ~ B if and only if An B = A.

Introduction to Real Analysis 3
Prove Theorem 4.3.11.

Introduction to Real Analysis 3
The first few terms of a sequence (xn) are given below. Assuming that the "natural pattern" indicated by these terms persists, give a formula for the nth term Xn' (a) 5,7,9,11,"" (b) 1/2, -1/4, 118, -1116"", (c) 1/2,2/3,3/4,4/5"", (d) 1,4,9, 16, ....

Introduction to Real Analysis 3
The sequence (xn.) is defined by the following formulas fo.r the nth term. Write the first five terms in each case: (a) xn:= 1+ (-It, (b) xn:= (_I)n In, 1 1 (c) x := 1 \ ' (d) X:= _2 . n _1- I n + 2

Introduction to Real Analysis 3
Determine a condition on lx - 11 that will assure that: (a) lx 2 - 11 < 4, (b) lx 2 - 11 < 1/10-3 , (c) lx 2 - 11 < 1/n for a given n eN, (d) lx3 - 11 < 1/n for a given n eN.

Introduction to Real Analysis 3
Establish the Discontinuity Criterion 5.1.4.

Introduction to Real Analysis 3
List the first five terms of the following inductively defined sequences. (a) xI:= 1, xn+1= 3xn + 1, '. I (b) YI:= 2, Yn+1= 2"(Yn+ 2IYn), (c) ZI:= 1, ~:= 2, zn+2:= (zn+1+ zn)/(zn+1- zn)' (d) sl = 3, s2:= 5, sn+2:= sn + sn+I'

Introduction to Real Analysis 3
For xn given by the following formulas, establish either the convergence or the divergence of the sequence X = (xn). n (_I)n n (a) xn:= n + l' (b) xn:= n + 1 ' n2 2n2 + 3 (c) x := -, (d) x:= ~ . n n+l n

Introduction to Real Analysis 3
Exhibit a bijection between N and a proper subset of itself.

Introduction to Real Analysis 3
Apply Theorem 4.2.4 to determine the following limits: (a) lim (x + 1) (2x + 3) (x e R), (b) lim x 2 + 2 (x > 0), x-1 .x--.1 x 2 - 2 (c) ~ C : 1 - ~) (x > 0), (d) ~ : 2 : ~ (x e R).

Introduction to Real Analysis 3
Prove the Sequential Criterion 5.1.3.

Introduction to Real Analysis 3
Complete the proof of the Density Theorem 2.4.8 by removing the assumption that x > O. 1

Introduction to Real Analysis 3
Give an example of two divergent sequences X and Y such that: ) (a) their sum X + Y converges, (b) their product XY converges.

Introduction to Real Analysis 3
Let Sl := {x e IR: x ::: OJ;Show in detail that the set Sl has lower bounds, but no upper bounds. Show that inf Sl = O.

Introduction to Real Analysis 3
Showthat lim(xf(x + n = 0 for all x e JR,x 2: O.

Introduction to Real Analysis 3
Show that the polynomial p(x) := x 4 + 7x 3 - 9 has at least two real roots. Use a calculator to locate these roots to within two decimal places.

Introduction to Real Analysis 3
If I := [0,4], calculate the norms of the following partitions: (a) 1'.:=(0,1,2,4), (b) 1'2:=(0,2,3,4), (c) l' 3 := (0, 1, 1.5,2,3.4,4), (d) 1'4:= (0, .5,2.5,3.5,4).

Introduction to Real Analysis 3
Prove Theorem 4.3.2.

Introduction to Real Analysis 3
Solve the following equations, justifying each step by referring to an appropriate property or theorem. (a) 2x + 5 =8, (b) x2 = 2x, (c) x2 - 1 =3, (d) (x - l)(x + 2) =O.