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!

Homework Help: A simple question about convergence

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Let x_n be a convergent sequence with a ≤ x_n for every n, where a is any number. Prove that a ≤ lim x_n when n→∞.

    2. Relevant equations

    Definition of limit. The usual ε, N stuff.

    3. The attempt at a solution

    Let lim x_n = x and choose ε=x_n-a. Hence we have |x_n - x| < x_n - a which shows that -x<-a and thus x>a.

    Is this valid?
     
  2. jcsd
  3. Nov 3, 2012 #2
    You do realize that [itex]\varepsilon[/itex] depends on n?? For a different n, you'll get a different [itex]\epsilon[/itex]. Do you really want that??
    Also, why is [itex]\varepsilon>0[/itex]? Specifically, why is it nonzero?
     
  4. Nov 3, 2012 #3

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi bedi! :smile:

    (try using the X2 button just above the Reply box :wink:)
    but ε has to be independent of n :confused:

    (ooh, micromass beat me to it! :biggrin:)
     
  5. Nov 3, 2012 #4
    Yes, you are right. However I still can't see the solution :(
     
  6. Nov 3, 2012 #5

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    try assuming the contrary :wink:
     
  7. Nov 3, 2012 #6
    Alright, so this will imply that x_n converges to both a and x which is a contradiction. Am I right?
     
  8. Nov 3, 2012 #7
    Take [itex]x_n=2[/itex] for all n and take [itex]a=0[/itex]. Then certainly [itex]x_n[/itex] does not converge to a. So no, you're not right.
     
  9. Nov 3, 2012 #8
    But I assumed that a>x. So -a<-x and x_n-a<x_n-x<ε. Hence x_n-a<ε ?
     
  10. Nov 3, 2012 #9
    Although the idea is there, the proof is still not very nice. For example, what is [itex]\varepsilon[/itex]?? You got to say things like this, not just introduce them without telling anybody what it is.
     
  11. Nov 3, 2012 #10

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    bedi, you know this is supposed to be a delta,epsilon proof …

    so you must define delta, and you must define epsilon​
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook