MATH 210 Exam 1, Fall 2010

Math 210 - Quiz 1 (Fall 2010) T. Tlas 1. (10 points each) Prove the following claims: i- {an}, {bn}S21 are two sequences converging to L and M respectively then an + bn — L-+M. Moreover, show that the conclusion holds even if we allow L and M to take the values -+oo and —oo as long as L + M is not of the form oo — oo. ii- Prove that 2 — + as long as L # 0. Moreover, show that if we interpret & and & as equal to 0 the result still holds. iii. Give an example of a sequence {an}%2, such that an — 0 but 2 -» 00 or - —o. n in re 2. (10 points) Can there be a cauchy sequence of rationals which does not converge to a rational? Justify your answer. 3. (6 points) Prove that 3°) i, converges. 4. (10 points each) Answer, with justification, the following questions: i- Is there a sequence whose range has no limit points? ii: Is there a convergent sequence whose range has no limit points? iii- Is there a cauchy sequence whose range has no limit points? iv- Is there an unbounded sequence whose range does have a limit point? 5. (7 points each) Give an example of a collection of open sets {O, }92, such that i- (R21 On is not open. ii- (821 On is not closed.