Understanding Analysis

Stephen Abbott, Middlebury College

Springer-Verlag Press, New York, 2001

Corrections to the text:

page 17, Ex 1.3.3(b): greatest upper bound should be greatest lower bound

page 24, proof of Thm 1.4.11 (a): The set A_1 should be defined
as a special case.

page 29: Exercise 1.4.13 assumes that X and Y are disjoint.

page 34, line 22. R. Beurkel points out that only relative consistency is known.

page 83, exercise 3.2.5. The  last expression is: Ve(x) cap A={a} when it should read Ve(a) cap A={a} instead.

page 107: last line of Corollary 4.2.5. The statement lim_(x->c) f(c) should have f(x) in place of f(c)

page 113: The answer to Exercise 4.3.8(b) would be "no" as written. However, the hypothesis of continuity (which is included in (a) and was intended to be included in (b)) would make the answer "yes."

page 114, Definition 4.4.1. says f is bounded in sense of definition 2.3.1,  but this should be definition 3.3.3.

pg 117, Theorem 4.4.6 is certainly true but could be an "if and only if" statement

page 122: In the first paragraph following the heading "Completeness," the point c should be in the interval (1,2) and not (0,1).

page 126: line 5, one of these limits should be a left hand limit and the other a right hand limit.

page 188: Definition 7.2.5 should define the lower integral L(f) as the supremum of the lower sums L(f,P)

page 187: In the proof of lemma 7.2.3 in the equation line, the expression m_k(x_x ... should be x_k and not x_x

page 187: In the proof of Lemma 7.2.4 the containments Q subset of P_1  and Q subset of P_2 should be the other way around..

page 195: 13 lines from the bottom in proof of Theorem 7.4.1, U(f,P) - L(f,P) should be U(f,P_2) - L(f,P_2)

page 199: The statement of Exercise 7.4.7 is incorrect and the example at the top of the preceding page is a counter example. The functions g_n need to be uniformly bounded.

page 209: R. Beurkel suggests K.R. Stromberg, An Introduction to Classical Real Analysis, as a good reference for the non-trivial details requested in Exercise 7.6.17. As he rightly points out, the piecing together done above must also enable this inequality to hold.

page 220: line 9 from the bottom: c should be c_k

page 220: line 4 from the bottom, x(3/2) should be x^(3/2)

page 222: Exercise 8.2.3 on the discrete metric was added late and actually duplicates other exercises (such as 8.2.1(b))

page 227: line 7 from the bottom, x should be x_k

pg 232: in the last line: sin(nx) sin(nx) should be sin(mx) sin(nx).

page 241: line 9 from the bottom, S_N should be S_n






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