Real Analysis Book 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 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".