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Monotone limit at a and b

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data
    let f: [a,b] ---> R be monotone. Prove that f has a limit at a and b.


    2. Relevant equations



    3. The attempt at a solution
    here is my proof:
    We need to show that f has a limit at a and b.
    Consider f:[a,b]--> R where f is increasing. Now for all x elements of [a,b] f(a)<f(x)<f(b), hence f is bounded and by the monotonocity, f cannot be oscillatory. Suppose x1<x2<...xk. Then f(a)<f(x1)<f(x2)....<f(xk)<f(b). So limx-->xk f(x)=f(xk)<f(b)and f has a limit at b. Then Limx-->x1 f(x)=f(x1)>f(a) and f has a limit at a.


    I'm not so sure about this one...
     
  2. jcsd
  3. Nov 13, 2008 #2
    Me neither.

    What you want to show is that if you have a sequence [itex](x_k)_{k\geq 0}[/itex] with [itex]x_k\to b[/itex] then the sequence [itex](f(x_k))_{k\geq 0}[/itex] converges. This sequence does not need to converge to f(b) which would mean that f is continuous at b - a fact which might not generally be true and which you are not asked to prove.

    As for what you have done so far: I can see only few things there which are correct.

    You said [itex]\lim_{x\to x_k}f(x)=f(x_k)[/itex]. This in general not true unless f is continuous at [itex]x_k[/itex]. Moreover you conclude from this (erroneous) statement that f should have a limit at b. How is the behaviour of f at [itex]x_k[/itex] related to the behaviour of f at b? - Not at all.

    I suggest you give it another try:smile:
     
  4. Nov 14, 2008 #3
    you can define (why?) A = sup{f(x)|a<=x<b}, which is the key to this question IMO, the rest of the proof becomes easy: prove such A is exactly f(b-) by a simple "epsilon-delta" argument
     
  5. Nov 15, 2008 #4
    [itex]\lim_{x\to x_k}f(x)=f(x_k)[/itex]. I'm still not quite sure what to do here instead.
     
  6. Nov 15, 2008 #5

    Office_Shredder

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    If you have a sequence (xk) that approaches b on the left, then assuming xk increases monotonically, you have that f(xk is increasing and bounded above. Hence?

    Then see what happens if different sequences have different limits
     
  7. Nov 15, 2008 #6
    Where did you get the approaching b from the left? Because of the fact that we have a<x1< x2....<xk<b?
     
  8. Nov 15, 2008 #7

    Office_Shredder

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    The question is prove f has a limit at b. It makes sense to pick an arbitrary sequence that approaches b to investigate what the limit would be (often, instead of trying to prove that it as a limit, you start by finding what the limit should be, and then proving that is the limit).
     
  9. Nov 15, 2008 #8
    Ok, so because the sequence approaches b, then the limit will be b? Then, do the same with xk approaching a from the right?
     
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