Entire Book: [Incomplete Assumptions] Assume that when Rudin says metric space he actually means non-empty metric space; troubles exists for instance in Chapter 9 and the statement of the contraction mapping theorem [Jeremy Rouse]
Definition 1.5 (ii) If x,y,z are in S, if x < y and y < z, then x < z. [Amanda Jacob]
Theorem 1.21: [Incomplete Assumptions] reads For every real x > 0 and every integer n > 0, there is one and only one real y so that y^n = x; Surely he means nonnegative real y, for 2^2 = (-2)^2 = 4 [David Eger]
Definition 2.9. When defining the intersection over a collection of sets, one needs to require that the collection be non-empty. [Wolfgang Helbig]
Remark 3.36: [False] Rudin says "Whenever the root test is inconclusive, the ratio test is too." However, examining his version of the tests with a_n = 1, the root test is inconclusive, but the ratio test indicates divergence. [David Eger]
Exercise 5.13: Assuming f is supposed to be real-valued, f(x) is not defined for all real numbers a and x in [-1,0). (e.g. a=1/2). [Nick Harvey]
Theorem 6.16: In equations (24) and (26), 'i' should be 'n'. [Nick Harvey]