Math 301 Midterm Solutions

# Math 301 Midterm Solutions - Math 301/ENAS 513 Fall 2007...

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

Math 301/ENAS 513, Fall 2007 Midterm Solution Triet Le October 29, 2007 1. (10 points). Prove that each bounded real sequence has a convergent subsequence. Discuss how this result is also true for bounded complex sequences. You can assume the followings: i. Every monotone and bounded real sequence converges. ii. Every real Cauchy sequence converges. iii. Every subsequence of a convergent sequence converges. iv. z n = a n + ib n z = a + ib a n a and b n b . v. { x n } ⊂ R converges if and only if lim inf n →∞ x n = lim sup n →∞ x n . Solution: Let { x n } be a bounded real sequence. I.e. there exists and M > 0 such that | a n | ≤ M for all n . One way to to prove this result is to use the properties of lim inf n →∞ x n = a to construct a subsequence that converges to a . The same also holds for lim sup n →∞ x n = b . The following proof is using the divide and conquer method. Let a 0 = - M and b 0 = M . We have a 0 a n b 0 , for all n . We will construct a subsequence as follows: First, we divide the interval [ a 0 , b 0 ] into two intervals of equal lengths, [ a 0 , ( a 0 + b 0 ) / 2] and [( a 0 + b 0 ) / 2 , b 0 ]. Either of these two intervals must contain -many x n ’s. If [ a 0 , ( a 0 + b 0 ) / 2] contains -many x n ’s, then we let [ a 1 , b 1 ] = [ a 0 , ( a 0 + b 0 ) / 2]. Otherwise, let [ a 1 , b 1 ] = [( a 0 + b 0 ) / 2 , b 0 ]. Pick any x n 1 [ a 1 , b 1 ] ∩ { x n } . Note that | b 1 - a 1 | = ( b 0 - a 0 ) / 2. Suppose x n k [ a k , b k ] ∩ { x n } , for some k 1, is chosen with | b k - a k | = ( b 0 - a 0 ) / 2 k and [ a k , b k ] contains -many of x n ’s. Again, we divide [ a k , b k ] into two intervals of equal lengths, [ a k , ( b k + a k ) / 2] and [( a k + b k ) / 2 , b k ]. If [ a k , ( b k + a k ) / 2] contains -many of x n ’s, then we let [ a k +1 , b k +1 ] = [ a k , ( b k + a k ) / 2]. Otherwise, we let [ a k +1 , b k +1 ] = [( b k + a k ) / 2 , b k ]. We then pick x n k +1 [ a k +1 , b k +1 ] ∩ { x n } such that n k +1 > n k . And continue. Note that | b k +1 - a

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 ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern