MATH 210 Exam 2, Fall 2016

Math 210, Introduction to Analysis- Fall 2016-2017 Final Exam, December 14. Time 140 minutes ous points : 90. Budget your time wisely. uesti estion 1. (12 pts) X aes Let f : X —+Y be bijective (1-1 and onto) and X compact. Suppose f is continuous. a) Prove that for every V C X, we have (F(V))° = F(V)- b) Prove that f is an open map ( you can use without proof that i when f is continuous). c) Conclude that f-' : Y — X is continuous. fi (compact) js clo Quiestion 2. (12 points) . Let X =C({0,1],R) = {f : [0,1] >R; fis continuous: d(f,9) = {If = allio = sup Vi €[0,1} (1) Prove that d is a metric on X° (2) Prove that (X, d) is a complete metric space.” (3) Define T : X —+ R by T(f) = {(0). Let U is open in (X,d). . (4) Prove that the function #:U > Risco Question 3. (12 points) Consider a) Find wi Tersuctih . el yialass brief justification the radius of convergence of the followi ret vine Wore gaa b Re ) Determino if the following series converge (you may use the nan > Vine i+ (nl) pousstion 7. (12points) Let A and B be two subsets of R that are} and @ =sup B. Define C= A+B {2=04+4; ne Aandy € Bh a) Prove that sup C@ = a+ B. (You may use characterizatio b) Consider the set $= {GIE* + 7 °V6, for mn € N}. U mas + 2 sup Si ¢) Consider the sequence am = Garg: +3 "6. Define lim sup with I brief justification. (Be careful not to mix SUP Zn Question 8. (12 points) a) Pind using the definition th f(z) = ve cae oi b) (UNRELATED)Prove the following inequalities (you i, Va € (0, %),0 <sinz <& < tang / ii. Va ve (0, Z),|tana — tan | >|a—-dl. P he derivative of the function”