Real Analysis: Errata

Major Errors

Page 12, in the Order Axiom I should have specified that the positive elements are non-zero. That is, I do not want to include 0 in my set P.
Page 30, first paragraph after Definition 1.29, line 3: Letting a = (x+y)/sqrt(2) was incorrect. That number is not always between x and y. One option (for x < y) would be to use x + (y-x)/sqrt(2).
Page 38, exercise 1.25: Assume A is a set of natural numbers, so that “maximum” has a meaning.
Page 38, Exercise 1.26: Definition 1.19 only defined completeness for ordered fields. But N is not a field. Oops. Either way, as an exercise you can show that N has the least upper bound property.
Page 84, at the end of the paragraph is max{a_1,a_2,…,a_n,a+1}. That “a_n” should have been “a_N”.
Page 86, second to last line of the Scratch Work: ε/2a should have the a in absolute value bars. That is, it should be ε/2|a|.
Page 92, line 5: α – ε < aₙ < α should be α – ε < aₙ ≤ α.
Page 107, second paragraph of the proof, line 2: Lemma 3.42 should be Lemma 3.41.
Page 129, the last equation in the block of equations as well as the first line after this block: there is a 1/2ᵖ term, but the p should be a p-1.
Page 130, the inequality just above the last paragraph: a₃ – a₄ should be a₄ – a₅.
Page 130, the final equation: at the end, the a_{2n} should be -a_{2n}.
Page 140, Exercise 4.1, part (e): Each n should be a k.
Page 141, Exercise 4.9, line 2: the r should be in absolute value bars, |r|.
Page 146, Exercise 4.29 part (b), the inequality 1/(n+1) < p/q < 1/n should be 1/n < p/q < 1/(n-1).
Page 167, Exercise 5.27, the set B should have been A.
Page 182, the line right about the Solution: the limit should go to 2, not to 3.
Page 194, the last line of the proof, V_epsilon(x_0) should be V_epsilon(f(x_0)).
Page 196, the second to last line in the first proof on that page: Uₖ should be Uₙ.
Page 196, the proof at the bottom of the page, line 1: Theorem 6.29 should be Theorem 6.30.
Page 266: In the two bullet points, (i-1)³ and i³ should be (i-1)² and i². Likewise in the next equation line.
Page 291: There is an issue with either the summation or the definition of the x_i. Each of the N+1 rectangles has a left endpoint and a right endpoint, so there should be 2(N+1) endpoints, not N+1. Probably the easiest fix is to say that the kth rectangle has endpoints x_{2k-1} and x_{2k}. Then the sum can still run from i=1 to i=N+1, provided we swap out each (x_i – x_{i-1}) for a (x_{2i} – x_{2i-1}).
Page 320, Exercise 8.44: The two terms on the right side of the inequality should have been squared.
Page 355, the second summation: The random minus sign should be a plus sign.

Minor Errors

Page 28 and elsewhere, the “Archimedean principle” should probably be “Archimedean property”
Page 35, Exercise 1.2, I called some natural numbers integers, which might be confusing.
Page 109, I say Exercises 3.7 and 3.8 are about limit laws, but it should be Exercises 3.8 and 3.13.
Page 110, Exercise 3.3, “give an example a sequence” should have been “given an example of a sequence”
Page 111, Exercise 3.10 should say “Prove that if” instead of “Assume that”.
Page 115, Exercise 3.40(a), “A Cauchy sequences” should be “A Cauchy sequence”
Page 121, in the paragraph after the proposition, in the last line “converges to 0” should just be “converges”
Page 163: Fourth bullet has erroneous “m” in “Exercise 5.31m”. Fifth bullet reference to part (f) should be part (g).
Page 207: Second bullet, the second “6.24” should be “6.25”.
Page 209: Exercise 6.11 has an erroneous “for”.
Page 212, Exercise 6.30: It is actually unnecessary to assume that f is continuous. I could have simply said it was a function.
Page 212 and 214, Exercises 6.32 and 6.50 are identical.
Page 215, Exercise 6.56, line 2: It probably would have been better to state that C is from the interval [0,1), rather than say only that C < 1 and leave it to the reader to deduce that C must be non-negative.
Page 216: Exercise 6.58e has a misplaced }.
On page 228, in the Proof idea’s second paragraph, I claimed that f(x)g(x) is in the first numerator post-miracle, but it is actually f(x)g(c). I should have noted that another limit would be need to turn that g(x) into a g(c).
Page 247, last paragraph: both times I said “holds” it should be “hold”. Also, “we proven that” should be “we have proven that”.
Page 250, Exercise 7.4, “function” should have been “functions”
Page 252: Ignore Exercise 7.19.
Page 268, the list of types of integrals: Statonovish should be Stratonovich.
Page 317, Exercise 8.33(d): I should have called the constant γ instead of e, since this is not the much more familiar Euler’s number (e=2.718…)
Page 319, Exercise 8.39(d): In the parenthetical, “part (c)” should have been “part (b)”.
Page 363, Exercise 9.16, line 3: “than” should be “that”.