Practice Exam 1 - Analysis Solutions

CJ | ff[ I«U< IWU»« If} Amherst ° Department of Mathematics and Statistics COMPREHENSIVE EXAMINATION < ANALYSIS O PRACTICE EXAM 1 NUMBER : Read This First : • ^ > olv/ ^Ton S . This is a closed- book examination . No books, notes , cell phones , electronic devices of any sort , or other aids are permitted. Cell phones are to be silenced and out of sight . • Write your number ( not your name ) in the above space. • For any given problem , you may use the back of the previous page for scratch work . Put your final answers in the spaces provided . • Additional sheets of paper will be available if you need them. If you use an additional sheet , label it carefully and be sure to include your number . • In order to receive full credit on a problem, solution methods must be complete, logical and understandable. Show all your work , and justify your answers. • The Analysis Exam consists of Questions 1-4 that total to 100 points. For Department Use Only: GRADER # 1 : GRADER # 2: Analysis - 1 Practice Exam 1 ( a ) State the Axiom of Completeness. - W 10 OON A rf ^ a loo* e. V» o» rvcl ^ ko iW Y t*\ -kA ©P f \ ontiwp 'V 'i poo . ( b ) Let ( an ) be a sequence of real numbers. State the e- N definition of what it means for ( an ) to converge to a 6 R . A ( &0 ) Jtfy 'K.ACt V f 70 611Z \0 N ^ iitk ^ iVi.fc-V ( c ) Let ( an ) be an increasing sequence of real numbers and suppose that there exists a real number MeR such that an < M for all n. Use the Axiom of Completeness and the definition in part ( b ) to prove that the sequence ( an ) converges. LeV (O k (t 6 cyhte ^4K M lo iu Th « i / ^ . Tkn Vy\ > Al Y ft , S \ A- CJ^ v^Vln ^ U - . ' Lc. loiirn. r 'i A. V' \ / n. f \ 0 r> tlH ii + i op A £ 70 i: V *- . cUirft. 4Vt*V SI ^ (JC iWrv\ tn + aft Or l 1 <*- » - 4 = ^ J $ ' . L / f RAI nuwWri ' * Ao c 4 VM * / U 4 - > r A iWl e *> sYs R be a function. Using the e-6 definition , define what it means for to be continuous at c 6 A . C 3 7 ^ ' f £^ i ^ ^ I s'* & Sn *• £> V£ 70 if ^A / Iffy. ) - 4 *7) 1 L l +u.« - X A ( b ) Suppose that the functions f , g : A —> R are both continuous at c i. Prove using the above definition that the function h : A » R defined by h ( x ) = / (x ) + g ( x ) is continuous at c. — ^ M H W Lt> i'V.af k ( H ‘{ A ^ 'P- ' CflD (XU cVoose. £ 7 0 » k S * TKJLV\ , 3 f*UL C\ Ky Sl oct ^ rvAvij TW »f cW X c. ( o b C OAA , ^ \>$. \ ov^Adt\ ^ ) ^ l A. cWeA . b rl | for " . K- c \odt- ^ \ ^ K. ~ o v t\ \ . l ^ \{ |\A dh & J W -tAU. La JBgU9 0\ _ ‘ CO (A (P etX Page 3 of 4 P ’ ho 4 V\ dAU bow ^ cl ^Ji . • U-* O VftV<- 3 O MA woiSJ ^v'A'i SiU\ ct nA^A vc > " (c ) Give an example to show that the union | I 1. (a ) Prove that ( /,, ) converges pointwise to a function / on [0 , 1] . \ if X 0 ( £ ,» ) COAVtr We tUuv\ 0 if b - \ , = T 0 - 4 ( ^ , If n( o) 4L ply 1 V \ -i 4 £ , ) < 6>"| < fjl , W « -W |i + ’-y \ B *V 11-|) = 0 , v. «L Wt n 1 1 + ha> i u «, Vx * O , A / 1 v \ *& v r\ 7/ N ; \ 4- n (i0 ‘‘ t * f ot rvVv< se. \ ( b ) Prove that ( /„ ) does not converge uniformly on [0 , 1] s OS ( * VWV Vn AcV \ oO yVN ^V fK • ^ , ^ 1 ^ e A WATWWA ^ on L ^ Q invow C0 AMer \ 6 l\ . 4 COA C° I i A ^ , ^ ConVinvo^ Vi v t\ \ . f ) JV f r0 <* if ^ o 4 M = Ql \ V 0x ir x - \ / VfcVu <-\l Vt our + (K i \ TWA •SVAC?. -+\$ X \ miA C\ cccVk 10 r> of - \ \ s TV"» i , C CAMMr\ oi ) i a\ l V -H + v poM * Ut Vo cj v i V"\ o \ . i\vfo /rr\ \ * Page 4 of 4 ^ ^ OA tooVinvo ^J tovQ . X 0 “