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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
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\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+dandac=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 al =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) = ((11)) + ((10) 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
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“12*“? '5— 5)“ + O 4 (9 + éb>> ma) L43] C37 "*5 C Cb+fb)] 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""“
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