M361K - Intro to Real Analysis Coverage: Second half of Chapter 4 (Exercise 4.4 onward), Chapter 5, Chapter 6 Definitions/concepts to Know: D.1: Convergence (of sequences): Define what it means to say\"{an } converges to c\" D.2: Cauchy Sequence: Let {an } be a Cauchy sequence. What do you know about it? D.3: Define a Deleted neighborhood and an accumulation point. What is the relationship between the two? What is the set of accumulation point of the interval (0, ) (, 4] R? D.4: Limit Definition for Functions: D.5: Open Cover: Provide a formal definition and an example of an open cover for (0, 1] R. Let's be cool and give an example of an open cover with no finite subcover. D.6: State the Heine-Borel Theorem D.7: Let f : D R be a function, c D. Define the following: f is continuous at c, f is continuous, and f is uniformly continuous. 1 Review Exercises 1. Prove that {an } = \u001a 1 1 1 1, , , , . . . 2 3 4 \u001b is Cauchy. 2. Prove the following: a.) For {an } as above, compute lim an . n Note: Recall the way we used Theorem C4 for stuff like this. Do we need it here? b.) If f (x) = 2x2 3x + 1 , then lim f (x) = 1 x1 x1 2 \u0013 \u001b \u001a\u0012 1 1 | k Z . Is this an open cover for Z? Is it possible to find a finite 3. Let C = k , k+ 3 3 subcollection of C that covers Z? 4. Is the following function continuous over R? Over Q? ( 0 : xR\\Q f (x) = 2 x : xQ 5. Let f and g be continuous functions from D R. Suppose f and g are continuous at c D. Prove that f + g is continuous at c. 3