1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Analysis: Least Upper Bound

  1. May 15, 2014 #1
    1. The problem statement, all variables and given/known data

    Find, with proof, the least upper bound of the set of real numbers E given by:
    E ={14n + 9/16n + 13: n [itex]\in[/itex] N}

    2. Relevant equations

    3. The attempt at a solution

    So I said that 16n+13>14n+9 for all N

    From this I get n>-2

    What do I do with this? I understand that as n increase E will decrease but I don't know how to answer the question. Any help would be much appreciated.
  2. jcsd
  3. May 15, 2014 #2


    User Avatar
    Homework Helper

    As written you have [tex]
    E = \{ 14n + \frac{9}{16n} + 13 : n \in \mathbb{N}\}
    [/tex] which has no upper bound.

    I infer from this that you actually meant [tex]
    E = \{ \frac{14n + 9}{16n + 13} : n \in \mathbb{N} \}.

    You should get that 1 is an upper bound for E. But is it the least?

    Let [tex]
    y = \frac{14n + 9}{16n + 13}
    [/tex] and solve for [itex]n[/itex]. You need [itex]n[/itex] to be positive, so that gives you a bound [itex]y_0[/itex] on [itex]y[/itex].

    You then need to see whether you can make [tex]y_0 - \frac{14n + 9}{16n + 13}[/tex] arbitrarily small by suitable choice of [itex]n[/itex].
  4. May 15, 2014 #3
    Hey pasmith,

    You're right, that is what I meant (long day).

    y = (14n + 9)/(16n + 13)

    And you said solve for n so:

    n= (9-13y)/(16y-14)

    However n would not be postive here?
    Last edited: May 15, 2014
  5. May 15, 2014 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Does N mean all integers (negative and positive) or just the non-negative positive integers?
  6. May 16, 2014 #5
    Hey Ray Vickson, N is natural numbers so all positive integers.
  7. May 16, 2014 #6


    User Avatar
    Homework Helper

    You can see that the numerator is positive when [itex]y < 9/13[/itex] and negative when [itex]y > 9/13[/itex] and the denominator is positive when [itex]y > 16/14[/itex] and negative when [itex]y < 16/14[/itex]. For the quotient to be positive, the numerator and denominator must have the same sign.
  8. May 16, 2014 #7


    User Avatar
    Staff Emeritus
    Science Advisor

    I don't see any point in solving for n. You are concerned with values of the fraction, not values of n. Instead, divide both numerator and denominator by n:
    [tex]\frac{14+ \frac{9}{n}}{16+ \frac{13}{n}}[/tex]
    Now, it is obvious what happens as n goes to infinity. Is y ever larger than that number for finite n?
  9. May 16, 2014 #8
    So the limit is 14/16 and as n goes to infinity.

    As n increases E decreases so away from L so its divergent?
    Last edited: May 16, 2014
  10. May 16, 2014 #9


    User Avatar
    Staff Emeritus
    Science Advisor

    I am no longer sure what you are talking about. Saying "the limit is 14/16" means the sequence is convergent, doesn't it? It has a limit. Clearly the fraction approaches 14/16= 7/8 as closely as we please. The only question left is "is it ever, for some finite value of n, larger than 7/8:

    Can you solve [itex]\frac{14n+ 9}{16n+ 13}> \frac{7}{8}[/itex]?
  11. May 16, 2014 #10
    Ok I understand that bit now.

    Solving the inequality:


    Which doesn't make sense so 7/8 is the least upper bound?
  12. May 16, 2014 #11


    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, trying to solve [itex]\frac{14n+ 9}{16n+ 13}> \frac{7}{8}[/itex] leads to a statement that is false for all n (I would not say "doesn't makes sense"- just "false") so the inequality is never true- 7/8 is an upper bound on the fraction. But we also know that it comes arbitrarily close to 7/8 so 7/8 is the "least upper bound".
  13. May 16, 2014 #12

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Alternatively, use calculus to examine the behavior of the function
    [tex] f(x) = \frac{9 + 14x}{13+16x}, \; x \geq 0[/tex]
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Analysis: Least Upper Bound
  1. Least Upper Bounds (Replies: 8)