MATH 210 Exam 2, Fall 2017

MATH 210: Introduction to Analysis Fall 2017-2018, Midterm 2, Duration: 60 min. Exercise | Points | Scores 1 40 2, 10 3 15 4 20 5 25: Total 110 INSTRUCTIONS: (a) Explain your answers precisely and clearly to ensure full credit. (b) No book. No notes. No calculators. CHEAT SHEET: (a) Let (X, d) be a metric space. A function f : X > R is Lipschitz if there exists a constant C > 0 such that for all x,y € X we have [f(2) — FY) < Cd(z, y). (b) Recall that d,, is defined on R2 as dqo(x, y) = max{|z; mae nil, |t2 — yol}, fore = (21,202) € R? and y = (ys, 42) € R2 (c) A function f : R > R is onto if for every y € R there is x € R such that f (xz) =. Exe ise 1. Warning: (a), (b) and (c) are in independen See 2G) ree tie tS d) be ametric: space. Stat ——— n_of convergen: ce of a sequence {xp} of elements of X. eta (b) (10 points) Prov © that £2 (a »bn)} 60 erges to (a,b) € R? for dy, if and only if HD fan} and {0} respec SG ND Ne D\ lt Zo Te. a a eles hex [ ye, het [a ra . : iL < “The fom a = hie ey a ler Slee, DS the Ms nS ay [ba- bes S fs n> nx my n-) : la-«/<f al et ke nae bleed [be hay Ae 4 A yo (Ces, N)) (4 ))) 4 (c) Let (X, d) be a metric space. Let K be a nonempty compact subset of X and let 2 € X. Define. d(xo one ae 1y) ly € Ky}. (\g0 a nts) Show that the fun , 2) defined on X is Lipschitz AC Xo, ) « den, y 4 } + dey x) if) eens na < dy, x) lier = ¢ 2) Uppos __|eeny + ; | ke (5) Bee 1. (5 points) Show that there exists y € K such that (zy) = eddy an AMC, =) kK —> “10 AR G2er G@ wna a ee Ate ‘4 Bee Ue ae 14 | Hox Jk} - | (xo) a A 5) Exercise 2. ——____——a)-(5-points)-Let 5~a;-be-aseries of of lim to show that >d a diverges. "ae py aS an’! fl pty A an ADO Z) Le Ay positive terms. Assume that lim(a,)2 >1.Use-a property b Q) (b) (5 points) Use the previous question to prove that ye. Qn With an = (2 + sin “ 1 r” diverges ifr >. Vole n° 1S eter 2 Ora), Sr os( NJe ae fol, im } f— 4. 26 iz CEO PS ZY, A Wo, Exercise 3. —_____(a)_Let }¢.a,, be an absolutely convergent series-and_ en -be-a-Lipsehitz function: i. GS points) Assume that f(0) = 0. Prove that )~ f(a,) is absolutely convergent. ¥ax)| = [ fee) -fo| < as Bye ee! x few) cv abaehs If, ii. ( points) Assume that f(0) 4 0. Prove or disprove, using an explicit counterexample that > f (an) is convergent. Coorlererep le Lk fot 2 a- = “= (b) (S points) Find a convergent series }> a, and a Lipschitz function f : R + R with f (0) = such that )> f (an) is divergent. Take fogels|, Den = T to N3le Late G2 ape [ri -gif< Pgh ae Exercise 4. Consider the set jj A= {ever ss} 1 (a) (15 points) Show using two ee methods that a closed. ce 2 vey fL - aie po) tl fat nS cme. mee (x Waxy (b) (5 points) Prove or disprove that A is compact. Ie ey eamier ase Tid £0 Ad C A bok tol elie oo a ne - : Bie Exercise 5. ate a ee CE cite aula ee Velie is evoyh dey 4 (Sue rf fog: be] XD fre {es x | oe cous bot eee fejey (Sov baja Be savin thee 2 x at foe. (b) Let f: R-Rbea uch that era Cale limy +4400 f(z) = +00. i. 6 poi eo ae Rsv ch that f(z) > 0. er (yee) fs AM xé Q 4 he oe. L fo ee KA 0 Al hee ts See Wey ian: Sd. iii. (5 points) Show that there uch that f(x) > f (2x1) if |z| < R. 4 ee ween = a Goyal os er Lewes j Hat rz) { Se. al Slay ees Rd. iv. (5 points) Deduce that f Bee a global minimum, namely that ther that f(x) > f(zx*) for all (0%) =_ baa Y (G0), fas X= Mo % abel € “= X, % > ba le eee (oe axe fenr fo) a F