advertisement

#1. True or False: If false, give a counter-example/explain. a. Every infinite sequence of real numbers has a convergent subsequence. b. If {an} is subsequence of {bn} and {bn} is subsequence of {an}, then the two sequences are equal. c. If En is closed for each n , then En is closed. n 1 d. If {an} and {bn} are sequences such that both {an}and { an bn} diverge, then {bn} diverges. e. The sequence {an} converges iff an converges to zero. f. The set {0,2} has exactly two accumulation pts. g. There is an infinite set which has an infinite number of accumulation points such that the set and the set of accumulation points are disjoint. h. There is a countable set which has an uncountable number of accumulation points such that the set and the set of accumulation points are disjoint. 2. Define each term: a. Let S be a subset of the real numbers. Then S is closed if and only if b. Liminf an c. The sequence {an} is subsequence of {bn} if and only if d. A Cauchy Sequence e. Let A An open cover of A is 2n 2 n 2 #3. Prove (Using the -definition): Lim 2 2 n n n #4. Prove: If Ei is open for each i Ei is open. , then i 1 #5. Consider the set S of all sequences {an} such that {an} satisfies: i. For each n , an 1 or an 2; ii. If an 2, then an1 1. Determine whether S is countable or uncountable, and then prove your claim. #6. Prove that every Cauchy sequence is bounded using the definition of a Cauchy sequence.