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Midterm 2 Review the Exam Will Cover Chapters 4 (Functions Midterm 2 Review The exam will cover Chapters 4 (Functions, injections, surjections, bijections, cardinality of sets etc.) and 13 (sequences, convergence, bounded, monotone etc.). 1 As I stated in class, the first problem on the test will be to state the definition of a sequence (an)n=1 converging to a limit L 2 R. You should also be able to state the other definitions in these chapters, and the exam will also ask for some of these definitions. Here is a list of definitions you should be able to state: • injective function • surjective function • bijective function • cardinality of a finite set • countable set • composition of two functions • convergent sequence • bounded sequence • monotone sequence • supremum and infimum of a set/sequence You should also be able to produce examples of each of these objects, and also examples which have various combinations of these properties (e.g. a function which is injective but not surjective, or a sequence which is bounded but not monotone), as well as when there exist no examples (e.g. an unbounded convergent sequence, which cannot exist because every convergent sequence is bounded). For example, if X = f1; 2; 3g and Y = f4; 5; 6; 7g, then an example of a function f : X ! Y which is injective and not surjective would be to define f as follows: f(1) = 5, f(2) = 4, f(3) = 7. n For example, a sequence which is bounded but not monotone is an = (−1) . You should be able to prove that a given sequence converges, and good practice for this would be (−1)n 1 1 1 p1 showing that sequences like n , n2 , n3 , n , n all converge. Here is an example of a proof that 1 n2 converges to 0. Proof. We wish to show that given any > 0, we can find an N 2 N such that for every index 1 q 1 n ≥ N, we have n2 − 0 < . Let > 0 be fixed but arbitrary. Consider the number which is q 1 2 1 1 positive. Choose an N 2 such that N > . Then we also have that N > , and so 2 < . N N 1 1 1 Thus, whenever n ≥ N, we have 2 ≤ 2 < . So for all n ≥ N, we see that 2 − 0 < . n N n Here is another example of the sort of thing that could reasonably be asked on the test. 1 Suppose f : A ! B and g : B ! C are both surjective. Show that the composition g ◦ f : A ! C is surjective. Proof. We need to show that for every c 2 C, there exists an a 2 A such that g ◦ f(a) = c. Fix an element c 2 C. Because g is surjective, there exists some element b 2 B such that g(b) = c. Then because f is surjective, there exists an element a 2 A such that f(a) = b. For this a, we see that g(f(a)) = c, and so there exists an element a 2 A such that g ◦ f(a) = c. Though the exam will not ask for some of the longer proofs, it could ask for shorter ones, or for ideas from the longer ones. For example, the following key fact came up in a couple of proofs, and has a short proof that could come up. Suppose (an) is a bounded sequence, and let S = sup(an). Show that for every > 0, there is an element an such that S − < an ≤ S. Proof. By definition, S is an upper bound for the sequence, so for every n, an ≤ S. Suppose that the claim is false, i.e. suppose that there is an > 0 so that there are no elements of the sequence with S − < an ≤ S. But then, for every n, we have that an ≤ S − . Thus S − is an upper bound for the sequence, and it is less than S. This contradicts the assumption that S was the least upper bound, and the claim is proven. As another example, the proofs of 13.14 (limits of sequences are unique) and 13.17 (if a sequence is bounded, it cannot converge to a limit outside of the bounds) in the book, which we worked in class, are based on a similar idea. So those proofs could come up. 2.
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