MATH 301 Midterm Solutions

MATH 301 Midterm Solutions - Math 301 Midterm. Solutions 1....

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Math 301 Midterm. Solutions 1. (40 points) In your answers you should refer to de&nitions or theo- rems/lemmas. (a) Is there a bounded increasing sequence in R which is not Cauchy? If yes give an example, otherwise prove that there is no such ex- ample. (b) Consider the decimal expansion of = 3 : 141 592 654 ::: and form a sequence ( x n ) 1; then x 1 = 1 10 : Take the next two digits 41; then x 2 = 41 100 ; the next 3 digits are 592 so x 3 = 592 1000 and so on (then x n = abc:::z 10 n where abc:::z are the "next" n digits). Does ( x n ) have any convergent subsequence? Prove your answer. (c) Let K be some closed set in the metric space ( M;d ) and let a be some element of M with a 62 K: Prove that there is some ± > 0 so that D ( a;± ) \ K = ²: (d) Prove that \ 1 n =1 & 0 ; 1 n ± = ² . Which property of R are you using? Solution 1 Each part is 10 points (a) No, because a bounded increasing sequence in R is convergent by the Completeness axiom and any convergent sequence is Cauchy. (b) The sequence is bounded, all terms are in the interval
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MATH 301 Midterm Solutions - Math 301 Midterm. Solutions 1....

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