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# MAT370HW2 - Exam 1 Real Analysis 1 Each problem is worth 10...

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Unformatted text preview: Exam 1 Real Analysis 1 10/3/2008 Each problem is worth 10 points. Show adequate justiﬁcation for full credit. Don’t panic. 1. State the deﬁnition of the limit of a function ﬁx) as x approaches +oo. Lei Foe) haemdrim (10%"? am? D :‘S WW W» M Ma \wﬁ as x—aoo it EaMWLssQrw £20 El (1 mil (lumbar [H70 531'. iQA-LVE (gr mm MA MD ”y 2. a) State the deﬁnition of an oscillatory sequence. c1. §£CIMCL Eggs is Cabcia\\c~*¢:\$\§\\$c— Hr SUQ‘E: ﬁts“? comm?) or cilvercga. k: ’coo or "m ‘ b) Give an example of an oscillatory sequence. M Clix "-3 GU“ 3. 3.) Give an example of a function that converges to 5 as x approaches +00. {:00 t 5- '0 b) Give an example of a set with exactly two accumulation points. _ J- 19 {sea-m [41+ ' FM 7- n n {22% [m dd. A Z“ 7% 15; Ms «W [a s a 4L ed S=fznlnem§ m 9 has Jump OLCUMUWM pol-«\$5. / 4. State the Bolzano-Weierstrass Theorem for Sets. Ahi ihttvﬁxte. 56“ MV is boundﬁé M05 CC} \€.03l-' Onﬂ aeeum eds} {on 905,1 3. \ M / 5 Prove directly from the deﬁnition that lim C x: C a, where c is a real constant 11-96 0,353 9270 28-70 33;. anw Decal-8 :5 \Q‘Czd— 1-\ " a La- ZVQYJQ‘CSM. w Lav 8 E/ch mm Q its cg raoA guacam’x \r—ch Hm. an}, wet/Maw \x—edé?) ‘30 we. how-Q. \x—OA " (if/VJ— ; we, «(10%: lax -c;.o~\4i =3 \c.»\\x~0.\‘-& --:> \Lx-QOA‘E“ ioHX \ E 'Q 4 mg m. '3 mm m on um \x-a\‘§<c\ x450... taxman. Va>© 33—24“ at, “’3‘ Q‘s/m CDC” \qQCr-\—- QLXLE. and. Wu. P (be; Q ”(‘3 (”QMQKQA‘Q“ ﬂé/ 5% M( E 6. Suppose that f and g are functions with both having domain D g R. Prove that if lim f (x) A and lim g(x)= B then 11m( f g)(x)= A- B. x—>a Wait, +£m7‘ a {Mac/f be. 4’"? ﬁcCamv‘aﬂﬁbﬁ ”“471. rm” 5?”!C/L l;,¥goc):&/ 19X 270‘ 3 {>0 3 b. ﬁOé/Kwﬁ/(S any £60117 (700*3/éi" ' [900/ /5/é ?cx) 3/: L90 f @cxj‘L/g/d'é ’0’?" +HCX)[4€‘*/5’ L67“ £+)5/: Kg, A/ow «we! C((QO [tr/‘0” jg >C) 5.6 QM? 3} 5376? 9.13 ‘ % 04/Xﬁl43 +Xé~031>2709“5/42WH ".2 1’0 A K may!) WW 9W7) 4 35; +5; 2:. a ‘ km 9 lyf a w >696? 'W) 748 D 0&5,- 7. State and prove the Monotone Convergence Theorem (proof of either case is acceptable). 5mg, *- .3195- m éCCLU-Qﬂte. is monokone, Omsk bOw’lCEQCM mm {Jr Converges‘ QFODX“ I COQEI bi)\f\{.\/\ iqng 15 {ALTEEQCYLCS L61” g>O, b4: jag/1. we,“ v39. \(mco £335 is bounded 50 Pr W155i he“: CA [4‘15 \N “a (omffmlme’ss Axp“om_ T {5. Wow-b “Ml" 1L LAQQQF \oowné, L. J L~f1 g5“ wﬁ‘r w. ax \eosf ”We! bound so ZIWQAMCL. m4 &*>Lei’. Aka blame, Om “5 {V‘C‘Fecﬂiﬁﬂ v36. lumped CA“ )3 Q ¥ lww >m+ , A-dcldlowlquy 0‘“ < L+b 5/1“; L IS MVP?” boma so MW. £10m be no more pat‘n‘ﬁ 0F c», do“ 1 08“ at hoe COM‘ﬁAQ W656i ”m eiuc‘ It. {13; U33, Ma we Li 4 9w 4c“ 4 UL V/vx >n*‘ W“ 5% erm‘x‘vi xj L-z. “an <L+a :> "L4On’Léi, 2-27 Ion—L‘ ¢L_ \Lh>n—¥. “9); “\$5 is We éﬂkniwon 0L aanveggeo. so Since QM 9L>O EW’FQVA book Add: \m— L] 43’ VIA >m’* n 8. Using some or all of the axioms: (A1) (A2) (A3) (A4) (A5) (A6) (A7) (A8) (A9) (Closure) a + 3), db E IR for any a, b E R. Also, if a, b, c, d E R with a = b and c = d, then a+c=b+danda-c=b'd. (Commutative) a + b = b + a and db = In: for any a, b E R. (Associative) (a + b) + c = a + (b + c) and (a'b)'c = a'(b'c) for any a, b, c, E JR. (Additive identity) There exists a zero element in IR, denoted by 0, such that a + 0 = a for any a E R. (Additive inverse) For each a E R, there exists an element wa in R, such that a + (—a) = (Muiiipiicative identity) There exists an element in R, which we denote by 1, such that a-l =aforanyaElR. (Multiplicaiive inverse) For each a E R with a at 0, there exists an element in IR denoted by % or a", such that era" =1. (Distributive) a-(b + c) = ((1-1)) + ((1-0) for any a, b, c E IR. (Trichoiomy) For a, b E R, exactly one ofthe following is true: a = b, a < b, or a > 1). (A10) (Transiiive) For a, b E JR, ifa < b and b < c, then a < c. (All) Fora,b,c€lR,ifa<b,thena+c<b+c. (A12) Fora,b,cEIR,ifa<bandc>0,thenacébc. Prove that ifa, b E Rh then a < b if and only if—a > —!:I. Be explicit about which axioms you use. L7 ,n/e; [anag- <5 N544. (A 4:16: “3me [A 77:? #0 é) 611%ij (was?) [4“ {j ()4 trim) My] 4; D+C—e)4:b+(~a)+(~b) MN] 42;) 0 + C b)4 b+(r4)ﬂfb)) E43] Vﬁﬂ (,4? (”2) +0 4 b + (leeway?) £493; vii “12*“? '5— 5)“ + O 4 (9 + éb>> ma) L43] C37 "*5 C Cb+f-b)] 4041) E24 5/3 e5 w e Owe C r4733. 69 ~54 C~a>+© we] 4’3“? Hbéﬂ C244] 9. Show that if a sequence {an} diverges to —oo and there exists some n] such that for all n > nI we have an 2 b”, then the sequence {39"} must also diverge to —oo. fel' rﬂ<br be -3.‘:uev\--‘2-""“ imita- “£0135; -—a" El M‘ 5.1” QLﬂé/Ul 00:? Haw ’0va 3M err Cum“ {9w H mu. bdi’ll Jel' m: MQKZM J ”*3 Cwi we 5e+ bh< 0m < M Rae qll Ln< M PM \J M<o ,5 “3””: 30> Srnce bu CM a» HM<O INC Mme Skown Hum" 856“) cyb -O\[60‘ k heth‘ue <59 COM [barman Weerem.> 10. If the sequence {an} converges to a nonzero constant A and an at 0 for any n, prove that the 1 sequence {—} is bounded. a N Wei/é W! 2’9”? WWW’e'WM mﬁa/W W M {1? Whyéf, 164m Wzawﬁa/ 2%me Ade/M" 44'; ”may”, ’3’? Wei/ﬁn WWW 44'9“. ‘3 ...
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