MATH 301 Midterm Solutions

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online