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The Unique Limit of a Complex Function

  1. Apr 2, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained.

    2. Relevant equations

    0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .


    3. The attempt at a solution

    Let S ⊆ ℂ.

    We consider some L' ≠ L, a limit of the function f(z) as z→z0. Choose ε = 1/2|L-L'| to find δ1 > 0, δ2 > 0 such that

    z ∈ S, 0 < |z-z0| < δ1 implies |f(z) - L| < ε
    z ∈ S, 0 < |z-z0| < δ2 implies |f(z) - L'| < ε

    Note that z0 is a limit point, so there exists some z* ∈ S such that 0 < |z0 - z*| < min {δ1, δ2}. Then

    |L-L'| = |L-f(z*) + f(z*) - L'| ≤ |L - f(z*)| + |f(z*) - L'| < ε + ε.

    This is the part where I get confused. I don't see how we can say that |L - f(z*)| + |f(z*)- L'| < ε + ε.
     
  2. jcsd
  3. Apr 2, 2015 #2

    Mark44

    Staff: Mentor

    Both |L - f(z*)| and |f(z*) - L'| are smaller than ε, so their sum would be smaller than 2ε. Is that the part you're confused on?
    Or are you confused about how they went from |L-f(z*) + f(z*) - L'| to |L - f(z*)| + |f(z*) - L'|?
     
  4. Apr 2, 2015 #3
    That is the part I'm confused on. I realise that they must be smaller than ε, I just don't see how they are.
     
  5. Apr 2, 2015 #4

    Mark44

    Staff: Mentor

    Since both L and L' are assumed to be the limits, it has to be the case that |f(z*) - L| < ε and that |f(z*) - L'| < ε. Most of the limit proofs work backwards from this conclusion to determine what the δ needs to be.
     
  6. Apr 2, 2015 #5
    Sorry, I thought I'd deleted part of the quote. This is the part I'm confused about:
    I realise that they must be smaller than ε, I just don't see how they are.
     
  7. Apr 2, 2015 #6

    Mark44

    Staff: Mentor

    See my edited post. I changed it after I wrote it.
     
  8. Apr 2, 2015 #7
    I think this is where I'm confused: In the first part there is |L - f(z*)| < ε, whereas you've written that is has to be the case that |f(z*) - L| < ε. Are the two equivalent? I see now how |f(z*) - l'| < ε and |f(z*) - l'| < ε must be true as they come from the definition of the limit. However, we have |L - f(z*)|.
     
  9. Apr 2, 2015 #8

    Mark44

    Staff: Mentor

    The two are equal.
    It's always the case that |a - b| = |b - a|.
     
  10. Apr 2, 2015 #9
    Thanks very much for your help.
     
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