First we show that it is monotone in particular increasing This is done by

First we show that it is monotone in particular

This preview shows page 7 - 10 out of 10 pages.

First, we show that it is monotone, in particular, increasing. This is done by induction. The base case is to show that x 1 x 2 . Is this true? Let’s see: x 1 = 1 , x 2 = 1 + 1 = 2 > 1 . True. Next, we assume that x n x n +1 . 1 Hint: Try an alternating sequence, which alternates between positive and negative terms.
Image of page 7
8 DR. JULIE ROWLETT Then, we must show that x n +1 x n +2 . Well, let’s try: x n +2 = 1 + x n +1 , but x n +1 x n so 1 + x n +1 1 + x n = x n +1 . Follow the inequalities and you’ll see that x n +2 x n +1 . Next, we’ll show that the sequence is bounded. Look at x 1 . Well, clearly x 1 < 2 . Will this bound work for all the rest? Let’s try. x 2 = 1 + 1 < 2 . Since we’ve shown the base case ( n = 1) and the next case ( n = 2) , it makes sense to try to prove the bound holds for all the remaining terms by induction. So, assume that x n < 2 . Then, x n +1 = 1 + x n < 1 + 2 = 3 < 2 . So, by induction, we have shown that 2 is an upper bound for the sequence. Hence, by the monotone convergence theorem, the sequence converges. What is the limit? This is fun. If x n x where x is the limit, then, by homework problem 16.11, x n +1 x also. So, lim n →∞ x n = x = lim n →∞ x n +1 = lim n →∞ 1 + x n . Since on the far right side, the only term changing in the limit is x n x, by Theorem 17.1, x = 1 + x x 2 - x - 1 = 0 . The solutions to this equation are 1 - 5 2 and 1 + 5 2 . Since the terms in the sequence are bounded below by 1 , the limit is also bounded below by 1 , by Theorem 17.4. Hence, the correct limit is the positive root, and so lim n →∞ x n = 1 + 5 2 . 3. Cauchy Sequences Theorem 3.1. A sequence of real numbers converges if and only if it is a Cauchy sequence. Proof: Assume the sequence of real numbers converges { s n } n N converges to s. Then, let > 0 . There exists N N such that | s n - s | < 2 for all n N. Hence, for all n, m N, | s n - s m | ≤ | s n - s | + | s - s m | < 2 + 2 = . That is the definition of a Cauchy sequence (MEMORIZE)¿ Next, assume that the sequence of real numbers is Cauchy. One reason it must converge is by the construction of the real numbers as the set of all rationals and
Image of page 8
MATH 117 LECTURE NOTES FEBRUARY 17, 2009 9 limits of Cauchy sequences of rational numbers. Another way is the following. We’ll
Image of page 9
Image of page 10

You've reached the end of your free preview.

Want to read all 10 pages?

  • Fall '08
  • Akhmedov,A
  • Math, Mathematical analysis, Cauchy sequence, DR. JULIE ROWLETT

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes