Section 7 #1. Left and right inverses weren't specifically part of the course, but you can find some similar examples and practice problems on pp68-71. #2, #3. These are portions of Theorems 7.15 and 7.17. Parts of those theorems are proven in the book. The skills problems from section 7 also included some proofs from Theorem 7.15; solutions are posted online. Section 8 #1. This is Definition 8.14 on p85. #2. The set of natural numbers is denumerable; (0,1) is uncountable and is a possible answer for (b) -- you proved in the takehome problem that it has the same cardinality as the set of real numbers. Answers are posted online. #3. (a) is very similar to practice problem 8.17. The "stronger" statement referred to in (b) is Theorem 8.18. #4. [This statement was updated to say "If S is denumerable, then there exists a proper subset T of S such that S~T." This is a more general version of examples we've seen in class, like S = the set of integers and T = the set of even integers. It's exercise 8.10, which was assigned as a skills problem. A solution is posted online. Section 10 #1. This was done in lecture; the Section 10 notes are posted online. #2, #3: These are exercises 10.4, 10.5 in the book. If you'd like to practice with other problems that have solutions posted, any of the relevant skills problems would work: 10.6, 10.7, 11.7. Section 11 #1. This is Theorem 11.9(c) #2. This is Theorem 11.9(d), the triangle inequality. The generalized triangle inequality (with more than two numbers) was assigned as exercise 11.7 in this section and was mentioned as a practice problem for Section 10, since the proof is by induction. Section 12 #1. This is Definition 12.5 and Practice Problem 12.6 on pp120-121. #2. These examples (and/or similar ones) were done in lecture and skills problems. (Solutions for the set {1/n} were posted as solutions to exercises 12.3 and 12.4, for example.) #3. This is exercise 12.8 in the book, which was never specifically assigned. It gives you some practice with the definitions of inf and sup as the *greatest* lower bound and *least* upper bound of a set. Let us know if you have questions on this problem. Section 13 #1(a) {pi} has no interior; its set of boundary points is {pi} itself. It's closed and not open. #1(b) The set Q of rationals has no interior. Its boundary points are the entire set R of real numbers. Q is neither open nor closed as a subset of R. #1(c) This intersection is the empty set. The empty set has no interior, no boundary, and is both close and open. (It's "clopen.") #1(d) The interior of this set is (x,y). The boundary points are {x,y}. It's neither open nor closed. #2 and #3 are nice problems, but above and beyond what you'd have to worry about doing on the exam. #3 was covered by at least one TA, if not more, as an example in class.